This solar system contains four earths. These earths, however, are opposite from each other in relation to the sun and their moons are vastly different. Say the moons and earths produce the same amount of forces and keep a stable orbit, at the same orbital speed, in the same orbital plane. Would it be feasible that the flora in the one of the earths is vastly different from the others? Here's a rough sketch of what im trying to describe: (sun - yellow, earths - red, orbit - black)
Using animals because it's clearer (and relatively the same, evolution-wise), if you're talking like you'll have intelligent octopi on planet A and clever tool-using monkeys as dominant species on planet B, yes, definitely your flora (and fauna) will be different. Depending on the local conditions (and a good amount of chance, too!), some plants and animals will take the lead on others, which will create different "eras" on each planet and will make your species change over time.
Moreover, because every species will evolve on different planets, their DNA will change too. And even if life came from the same, foreign source (a.k.a. panspermia), at some points your plants and animals will diverge too much to say they are part of the same family, and they won't be able to reproduce between each other.
Of course, all the above is supposing that all 4 Earths manage to keep life. As far as we know it, life coming on a planet is an incredibly rare event, so having four planets on the same system having life is like throwing a billion-sided die 4 times and bet it will fall on a 1 every time. The probability is insanely low 🎲.
Something that can reduce the differences
There's a scientific debate on how far things would change if Earth changed just a bit. Or more. As far as I know it's not settled yet (and probably will never), but what has already been settled is that there's some sort of convergent evolution between species on Earth over time. Crabs is one of such instance : Overtime, the lower part of the body rolls back under the shell, going from a body akin to a lobster and going to a rounded one. This happens in order to protect it from predators and reduce energy consumption. This evolution happened on several far-related species, leading to something that can deceive an untrained eye (for instance these genuinely true crabs vs this not-so-true king crab).
This same phenomenon could happen independently on each planet, leading to similar-looking yet entirely different species. For instance, since your planets are under the same influence of the sun, photosynthesis is likely to occur on each on them. It's likely to come under widely different shades and intensity depending on what currently works on the planet, but you're quite likely to have it somewhere on each of them, since it provides an easy energy source.
One must recognize that four planets in the same orbit would be stable for at most a very short time. But even without perturbed orbits or collisions the planets would have different histories. One planet might experience a gamma-ray burst while its opposite is shielded, which makes a different history. If the Earth had been in a different location, then big, lumbering dinosaurs might rule the Earth and such creatures as elephants might have no chance. (Marine mammals might exist, but we wouldn't). An African elephant would be easy prey for T. Rex. The biggest mammalian predators would have been the size of domestic cats and rat terriers, which leaves obvious limitations.
Snakes, some of the most successful of all predators, would be rarer because they required the flourishing of mammals and birds for their success.
I spoke of elephants largely for their intelligence and ecological role. Elephants do much to create the savannahs of Africa and Asia and would do much the same in Australia and the Americas if they got there. The mammoths created the steppe-tundra of the subarctic zones, and in their absence we now have boreal forests. Another creature that shapes the environment is a large rodent, the beaver, which would have been easy prey for carnivorous dinosaurs.
Part One : To panspermia or not to panspermia, that is the question
Panspermia (from Ancient Greek πᾶν (pan) 'all ', and σπέρμα (sperma) 'seed') is the hypothesis, first proposed in the 5th century BC by the Greek philosopher Anaxagoras, that life exists throughout the Universe, distributed by space dust, meteoroids, asteroids, comets, and planetoids, as well as by spacecraft carrying unintended contamination by microorganisms. Panspermia is a fringe theory with little support amongst mainstream scientists. Critics argue that it does not answer the question of the origin of life but merely places it on another celestial body. It is also criticized because it cannot be tested experimentally.
It might be possible for a giant meteorid or asteroid to strike a planet and knock rocks off it which would travel around the solar system and eventually land on another planet in the same star system. In fact scientists believe that has happened and that rocks from other worlds have arrived on Earth.
A lunar meteorite is a meteorite that is known to have originated on the Moon.
As of July 2019, 371 lunar meteorites have been discovered, perhaps representing more than 30 separate meteorite falls (i.e., many of the stones are "paired" fragments of the same meteoroid). The total mass is more than 190 kilograms (420 lb). All lunar meteorites have been found in deserts; most have been found in Antarctica, northern Africa, and the Sultanate of Oman. None have yet been found in North America, South America, or Europe.
A Martian meteorite is a rock that formed on Mars, was ejected from the planet by an impact event, and traversed interplanetary space before landing on Earth as a meteorite. As of September 2020, 277 meteorites had been classified as Martian, less than half a percent of the 72,000 meteorites that have been classified. The largest complete, uncut Martian meteorite, Taoudenni 002, was recovered in Mali in early 2021. It weighs 14.5 kilograms (32 pounds) and is on display at the Maine Mineral & Gem Museum.
Several Martian meteorites have been found to contain what some think is evidence for fossilized Martian life forms.
Lithopanspermia, the transfer of organisms in rocks from one planet to another either through interplanetary or interstellar space, such as in comets or asteroids, remains speculative. A variant would be for organisms to travel between star systems on nomadic exoplanets or exomoons.
So some scientists have proposed that microscopic lifeforms in Earth rocks could have landed on Mars and become the ancestors of hypothetical Martian life that presumably is now extinct. And possibly microscopic lifeforms in Martian rocks might have landed on Earth and become the ancestors of all Earth life.
And it is theoretically possible that some microrganisms in rock could survive being ejected into space travelling for thousands of years at least through the vacuum of space, and the shock of crash landing on another planet.
So in the fictional solar system you desire, it would be theoretically possible for microscopic lifeforms to spread from one of your four planets to the other three, and thus that all lifeforms on the four planets would be distantly related. Of course the path of evolution on each planet would be different, even starting from the same set of microbes, and so the advanced multicelled plants and animals in each planet would be different from those on the other planets.
Or there might never have been any spread of microbes from one planet to another, and their lifeforms might be totally unrelated, and even more different from each other, especially in their biochemestries. For exmaple, all lifeforms from different planets might be mutually inedible.
Part Two: Orbits
Gravitational problems are quite simple to solve when they are two body problems, involving only two objects.
But if there is a system with three bodies, or more, it becomes more complex. Their orbits can't be solved by elegant equations but by brute force running of many calculations, which in recent decades has been helped by advanced computers and orbital calculation programs.
You may have heard of the idea of a Counter-Earth, a planet orbiting the Sun in the Earth's orbit but directly opposite to Earth. Such a planet would be hidden from Earth on the far side of the Sun. But perturbations from the gravitational attractions of other planets in the solar system would eventually change the orbit of the Counter-Earth so it would be visible from Earth.
Your star system would have two sets of planet and counter-planet. If the four planets are numbered sequentially in order around the star, planet 1 would be at 0 degrees, planet 2 would be at 90 degrees, planet 3 would be at 180 degrees, and planet 4 would be at 270 degrees.
So planet 1 and planet 3 would be counter-planets to each other, and Planet 2 and planet 4 would be counter-planets to each other.
And I expect that system would be more unstable than a system with a star, planet, and counter-planet. The planets would more out of their proper positions, and probably each one would fall into the star, or collide with another planet, or be ejected from the star system into interstellar space and become lifeless.
I think that the only way your system could remain stable for geological eras of time, time enough for interesting advanced lifeforms to evolve on the four planets, would be if a super advanced society used incredibly powerful technology to keep the four planets in their proper positions.
So if you can't use a star system exactly like your original design, you need to find out what star system design will be long term stable and also most similar to your original design.
There is a blog called PlanetPLanet.net by astrophysicist Sean Raymond. On there, one section is called The Utimate Solar System, where Raymond tries to design scienfifically plausible star systems with as many habitable planets as he can.
In a post called "The Ultimate Engineered Solar System" Raymond says that a paper by Smith and Lisseaur shows that a number of planets can share the same orbit around a star if they have the same mass and are equally spaced. Apparently such a system can be stable with between 7 and 42 planets in the ring.
As Raymond says:
I can only think of one way our 416-planet system could form. It must have been purposely engineered by a super-intelligent advanced civilization. I’m calling it the Ultimate Engineered Solar System.
And if a star system where the planets were created and moved into orbit by a super advanced civilization a can fit into your staory, your can use a star system which has at least one ring of 7 to 42 habitable palnets.
And you should see another post:
Where Raymond apparently found ways to have smaller numbers of planets share the same orbit.
So possibly you might want to ask Raymond if your set up with four planets space 90 degrees apart on the same orbit would have long term stabiity, and if not, what would be the closest thing to it that would have long term stability.
Part Three: Scale of science fiction hardness
That great time waster TV Tropes has a trope "Sliding Scale/Mohs Scale of Science Fiction Hardness". The higher the score on the scale, the more scientifically rigorous and plausible a story will be. So if you are content with a low score on that scale and a scientifically unrealistic and implausible story, you can arrange your planets any way you want - while realizing that it is possible that some ten-year-old children will read it and snear at you for your scientific illiteracy.
Quite certainly yes.
Assuming that your planets have stable orbits, and perfectly identical starting conditions, varying flora and fauna in my opinion is guaranteed.
Every one of the planets will have it's share of chance events, influencing everything on the planet in big or small ways.
Things like the dinosaur killer won't happen on all four planets, and if they did. they wouldn't happen in exactly the same state of evolution. So even if we assumed a more or less deterministic outcome of evolution (which doesn't seem sensible, but anyway), resetting most of evolution by help of a big rock will leave different ecological niches and different survivors to evolve into filling them.
After only a few such random interventions, chances are very high that very large parts of your flora and fauna are completely different between the planets. The only thing they will most likely have in common is the very basic "rules" of life. If all four started with carbon - and DNA-bases, those are not very likely to go away, although, if you wanted, you could come up with a second, concurring setup, that was never fit enough to outcompete the carbon-based things, but got lucky by some random event, which killed out enough of the carbon based lifeforms to create a nice big enough for those seconds to grow large and dominant.
Assuming that life does form on each planet independently, on its own (i.e., not a form of panspermia), and it forms on each of them during a concurrent time frame, there's no reason to expect the biology found on each planet would be even remotely similar to the others.
There may be some similarities due to the chemistry at that temperature. But everything else would diverge from the start. Things like cells and DNA, those are expressions of biology found on Earth. Each of the planets would have their own form.
Its possible you'd see some convergent evolution - creatures filling certain niches such as plants / fish / birds. But its also possible there'd simply be no similarity. You might have a wildly diverse population of trillions of creatures on one planet, and only a handful of massive and ancient colony creatures on the other.