Can a story make the reader find it plausible that two different races evolved to intelligent civilization on two different worlds in the same star system, extant at the same time?
I think so. Planets share material, and in the resent Cosmos remake Tyson explained that rock blasted from the surface during an impact served as a lifeboat during the late heavy bombardment, so life (or however it's gotten thus far) could wait out the re-melting of the planet's surface for a while.
… interplanetary transit system that sends rocks between the planets.
Such a meteorite can safely shelter microscopic cargo the seeds of life an interplanetary ark.
Most rocks are porous, full of tiny nooks and crannies, where life can stow away.
We know that some microbes can survive the hostile environment of space.
Take these guys, for instance.
These microbes spent a year and a half riding on the outside of the International Space Station, exposed to the extreme temperatures, vacuum, and radiation of space.
And some of them were still alive and kicking when they were brought back to Earth.
Even more astonishing are these creatures, awakened from a deathlike sleep of eight million years.
They were frozen in the Antarctic ice millions of years before our species even existed. And they're still alive.
If life can withstand the hardships of space and endure for millennia, then it could ride the natural interplanetary transit system from world to world.
It's a good bet that our microbial ancestors spent some time in space.
Why do we think so? The Earth is four-and-a-half- billion-years old.
For the first half of its lifetime, large asteroids were bombarding the planet every few million years.
The most violent impacts vaporized the oceans and even melted the surface rock.
Each such collision would have completely sterilized the planet for thousands of years.
But we know from fossils in the rocks that bacteria were evolving on Earth during this formative period.
So how could life have survived such a lethal series of blows? Whenever one of those big asteroids hit the Earth, the explosion would blast out a crater, launching millions of boulders into space.
Many of those rocks carried living bacteria inside.
Some of the bugs would have survived in space, while all those left behind on Earth would have been fried.
A few thousand years after each impact, the Earth would have cooled down enough for water to condense into oceans.
The planet would again be habitable.
Meanwhile, most of the rocks launched into space would have been orbiting the Sun.
Some of them would encounter the Earth again, reenter the atmosphere as meteorites, and deliver their precious cargo of life to re-seed the planet like Noah's ark.
What this means is that life doesn't have to start over again from scratch after each catastrophe.
It can pick up where it left off.
When the solar system was young, Venus was probably more like Earth, with oceans and maybe even life.
Venus, Earth, and Mars were all exchanging rocks with each other, due to asteroid impacts.
Does life on Earth carry any traces of interplanetary voyages made in the distant past? Why is it that some microbes can survive the intense radiation and vacuum of space? These conditions don't naturally exist on Earth.
Maybe those bugs are telling us that their ancestors survived those same conditions in space, a few billion years ago.
So we know that microbes can stow away in rocks and survive the voyage from planet to planet.
So, consider multiple planets in the habitable zone that were not fatally flawed and made it to the point of having a feedback system maintain a stable environment. They would all have biospheres, and the life would be related.
Suppose that major stressors such as impact events are also naturally correlated. Whatever disturbed the system such as a passing star or the sun passing through a dusty region of the galaxy, it would not be unreasonable for both planets to suffer impact events within a few million years of each other. Suppose that stresses like this are necessary for the biosphere to advance to more complex forms.
There are other stresses, such as major volcanism. It's a big galaxy, so why not? Statistically, the advancement of life follows an average time span for each step, so it's not unreasonable to suppose that both planets reached the kind of stage we are in, at the same time, within a hundred million years of each other. Perhaps the details are different in the final stages: we had an impact event wipe out the dinosaurs, but things were changing anyway due to (ultimately) plate tectonics. So maybe one planet has an impact and another has a ecosystem collapse due more to internal reasons from smaller outside stress. Maybe they had a different number of "repottings", but on average reached complexity ready to evolve intelligence at around the same time, because of the long time scale involved so individual events give way to long-term averages.
The stress that led to us was climate changes from forests to savana, etc. and eventually the ice age, and a genetic bottleneck that prevented evolving better fitness the old fashioned way. Suppose that both planets had complex advanced life and had been mature in that stage for a few million years. Some external stress, so it applies to both worlds, spurns development of intelligence. That way they'll be happening at the same time.
You can convince the knowledgeable reader to suspend disbelief that two intelligent races can emerge at the same time, within a finishing window of a few (single digit) million years. Why the final summit within a few thousand years? Perhaps ice ages, again correlated due to having external causes. At some point you have to appeal to "it's a big galaxy", but I think this is a platform that gets you most of the way there:
Development takes place, statistically, at a common rate. External stresses help synchronize things further.
Also, note that I said worlds, not planets. Perhaps they are moons of a giant planet, so they are more easily sharing material between them. Maybe the style of life on one takes spreads spores high into the air, so will naturally spread between worlds on rare cases, when those worlds are very near each other.