Lets pretend that Mars was a bit larger, so the gravity could hold on to the water and possible atmosphere, and then play along with the accidents that happened on earth happened at a similar tact.
Somewhere between when the telescopes appears and now we gaze at mars and see people just as us looking back.
What changes and impacts on space-faring would that make? The drive to meet those other people there.
EDIT: Removed irrelevant information.
First of all it would eliminate the question whether or not we are alone in space. Although, even with a very similar setup it is highly unlikely that two completely isolated species evolved in a similar time frame to similar extents of intelligence and such.
But ignoring that point: if in your setup this has happened, i think we (or they) would by now have built ships to go to mars. And long before that we will have tried to communicate with the Martians. We might have used something as simple as Morse signs with a big enough lamp.
I guess, if we do not take into account the possibility of sharing technologies via Morse code (or other communications technology), right after the moon missions the building of a mars going ship would have started, in even a bigger space race than the early cold war times brought forth, since prestige and potential profit from being the first to meet the Martians in person would have been huge.
Since all doubt of achieving anything "useful" would thus have been eliminated, it would have been almost trivial to convince the general public to cough up the tax money for that project.
Finally, the prospect of profit would most likely result in a higher risk tolerance than we encounter for current mars projects, which should again result in speeding up the projects.
Ok, I'll play, but I will also list a few reasons why this would be a wildly unlikely scenario at the end.
So, we have two sapient species, with early modern tech, finally able to see each other. With Human-level 17th century tech (first telescopes were developed in the early 1600s), you'd be able to get 30x magnification and see Sister-Mars as if it were the size of a grain of rice. I'll look at it from the Earth-centric perspective.
Look at this image from 10 feet (3 meters) away to get a sense of what early astronomers would see.
Of course, since Sister-Mars is double the radius of actual Mars, it would be double the size. Needless to say, you would not be seeing cities, or even mountains. However, you would see a blue and white jewel sitting in the dark sky. Thus, the existence of oceans and ice-caps might be quite easily resolved. Galileo's first book, the Sidereus Nuncius, would likely be about Sister-Mars and its oceans rather than Jupiter's moons. Oceans! On Mars! 20 editions would burn off the printing presses before the year is out. An enthusiastic Prince Maurits of the Netherlands offers a 2000 florins prize for an improved glass for the new farseeing telescopes, triggering an optics revolution. A jewish glassmaker in Amsterdam claims the prize, and mountains and vegetation cycles are soon noted for Sister-Mars.
A Spanish writer publishes a best-seller by the name of "The Ingenious Gentleman Don Quixote of La Mancha on Mars", where he depicts the hidalgo getting caught up in air vortices and dragged to Mars. By the end of the 17th century giant 300x telescopes are being built, and a thriving literary genre imagines various utopias and dystopias on Mars, inspiring generations of young budding scientists to dream the impossible. Disastrous attempts to launch projectiles "toward Mars" are conducted over and over again.
The intense attention to astronomy, and the craze among the nobility about court astronomers results in the laws of celestial mechanics being discovered a century early, triggering an early scientific revolution in Europe. A Dutch lensmaker by the name of Baruch Spinoza invents what we know as the Tsiolkovsky Rocket Equation, but means of propulsion into orbit remain beyond the era's technology.
A young Isaac Newton invents calculus and focuses on optics in an attempt to improve telescope resolution, discovering spectral lines in the process. Blaise Pascal devises means of calculating atmospheric interference with the telescopes, discovering many of the gas laws and pertinent information about atmospheric composition in the process, including many useful properties of steam.
Railway lines crisscross Europe by the mid-1700s, and a full blown industrial revolution is soon underway. By the early 1800s, Prussian artillery is finely crafted enough to launch projectiles on suborbital paths. Radio is discovered, and within 30 years, anomalous signals are detected coming from Mars. Carl August von Steinheil is the first to realize these were the product of another civilization.
The process of evolution works by selecting on the results of random variation from established forms, whereby the phenotype is exposed to the particulars of a time-space environment and successful reproduction in that environment allows the perpetuation of the genes in that genotype, even though the individual organism is generally not repeated. Reproductive success is therefore the product of numerous random mutagenic factors, as well as subject to exogenous shocks such as equilibrium-altering events (invasive species, climate fluctuations, massive ice ages, continent-wide volcanic eruptions and even massive space-origin impacts such as the one that is assumed to have killed off the dinosaurs).
Now imagine you have the extraordinary fortune to have not one but two Gigayear-habitable goldilocks-range worlds in the same system. By the established mechanism of asteroid exchange, life would most likely successfully spread to both worlds. So far so good. But from here on, life is set to diverge wildly. Even if the two worlds are similar as raindrops, there's a good chance you'll end up with trilobites or dinosaurs dominating on one world and mammals on another, since as far as we understand, many of the extinction events drastically altered global ecosystem balance, by literally wiping out the competition. The situation you describe would indeed be every xenobiologist's fantasy: how much convergent evolution does take place, and how much is dictated by random factors? Were dinosaurs doomed by their tiny brains, or would mammals have forever been stuck at the terrified night rodent stage if not for the Chicxulub event?
Ask yourself: What are the odds of two essentially independent planets happening to develop intelligent life at exactly the same time? Moreover, since humanity spent its first 100,000 years as hunter-gatherers, what are the odds the two independent sentient species would then proceed to develop agriculture at the same time? Yet more coincidentally, they also develop the scientific method within decades of each other. Far more likely that one would reach space tech while the other world has no sapiens-level intelligent life at all, or at the very least, while the other is stuck in its long hunter-gatherer dawn of civilization stage.
This can all be waved away with sufficient amounts of alien-ex-machina, who choose to come back every few million years and direct the evolution of life on both worlds, even transplanting individual genes, or perhaps full genotypes or phenotypes (that further their incredibly long term plans) from one world to the other. So I'll go ahead and assume that happened.
If you want to know more, you could to worse than to readi about the Drake Equation.
This reminds me of a series of short stories I read some years back where the premise was that interstellar space travel (FTL, via some kind of unexplained space warp) was incredibly easy, and that somehow Earthling technology just missed it, and went on to get to the cusp of interstellar travel the hard way, by actually flying through the intervening space.
As a result, humans eventually burst onto the scene of a community of traveling, trading sentients like the proverbial bats out of hell, with technology far more advanced than any other species. Most species (and this is why I was reminded of it,) were somewhere around the early Industrial Revolution, up until the late Victorian; so it was 21st century Earthling mech against aliens with steampunk tech.
The prequel to the series was The Road Not Taken by Harry Turtledove, and I'm pretty sure the collection was Herbig Haro. The same author also wrote A World of Difference which changes Mars to be the same size as Earth, and proceeds from there....
It sounds like you're driving at something similar, wanting to know what's the minimal level technology we'd need to start up interplanetary intercourse with our "sister planet." So here are some alternatives:
The age of Wooden Ships and Iron Men
Let's assume that sailing was the paradigm, and that solar sails could be made.* Well, few people realize that already in the 18th century submersible vessels were being built ... of wood! For example, The Turtle was an American submersible built to sink British warships at the beginning of the American Revolutionary War. (1776 - It worked as a submarine, but it failed to attach a mine that actually detonated a warship.)
The point is that similar technology could probably allow construction of solar sails that would spend months traveling the interplanetary void between Earth and Mars. And don't worry about them breathing - Antoine Lavoisier was synthesizing oxygen two years before the Turtle sailed in New York harbor. The problem is that getting up to space, and back down again is a darn sight trickier than traveling in space. The Robert brothers show you how to get part way there -- they built the first hydrogen balloon in 1783. And within two decades William Congreve was improving on Indian rocket technology for the British.
With a visible impetus during the preceding centuries to travel to Mars, I think it is not an unreasonable stretch to describe interplanetary sailing ships of combined wood and metal attempting the voyage by the mid 19th century, just as steam power and the Industrial Revolution are taking off. Hydrogen balloons would raise the people and goods to a certain height, and from there rockets would carry them the rest of the way to "floating docks" or "levitated islands" (space stations) from which they would embark on a solar sail journey to the other planet. I mean, I for one would willingly suspend my disbelief to a bit of plausible writing in an alternate history scenario like that.
* The big problem with making solar sails is making a thin enough, reflective enough fabric. Postulate an industry using spider silk to make the sails that developed a couple of centuries earlier.
Steam Power
(Sorry I haven't finished this yet.)
Since you've ruled the timeline to be irrelevant lets say the discovery happens around the late 19th century. Italian astronomer Giovanni Schiaparelli observed Mars in 1877 and described straight lines he called canali (sometimes translated as channels) on mars. Canal travel was huge at the time so this was taken as a sign of intelligent life like our own.
So in this scenario, instead of being proven wrong, he's eventually proven right. Albeit for the wrong reasons but that happens in science more than you might expect.
It took a fair amount of time to prove him wrong (some people still believe) but if you want to know what would happen if we thought there was life on mars, go back and read what did happen when many believed.
I think that jumping from the 13th - 16th century to space flight is a bit too broad, because in the middle several things might've gone differently.
For instance: Ever since the beginning of time, we've attributed the things we don't know how to explain to gods, so what if once we notice the martians, we take them to be those gods?
Then we (or at least a portion of the humans) would've grown to think of the martians as our gods, there would even be full blown riots from "martian worshippers" against the (unworthy) men of science trying to communicate with them. A sort of a holy war against scientists who claimed that "the gods from above are mere mortals like us".
Who knows, maybe space-related technology is eventually pushed back because of the power of those fanatics (which in this scenario would have proof that their gods are real, thus gaining far more adepts)
I'd like to think that eventually science men would prevail, but history would probably be very different and technology would take longer to develop because of the false belief on the martian gods.
Addendum
I want to also add what would happen once we do realize that the martians are just "people":
After that happens, all efforts would probably go into establishing a stable channel of communication, even before thinking about travelling there. And depending on their intentions (let's assume they're nice guys), our tech and knowledge would improve exponentially, just like theirs, since both civilizations would start sharing information and knowledge .
