Just pointing out some assumptions you're making and shoehorning in a couple of my own thoughts on the matter.

1. We're assuming that there is life elsewhere in the universe. (I personally tend to believe there is something, somewhere out there that could check the boxes to be considered "life" but it's not exactly a sure thing)

2. We're assuming that whatever life exists out there possesses a soul (If I did believe in souls, which I personally do not, I don't think that every living thing necessarily would have a soul- bacteria, fungus, plants, etc. I wouldn't think have souls, nor necessarily all animals, I don't think I'd say that things like placozoans, sea sponges, coral, and jellyfish have souls for example.)

3. We're assuming that those souls operate in a similar manner to our own and are compatible with us. As far as we can tell all life on earth evolved from a common ancestor, so all souls have at least have somewhat similar biological "hardware" (wetware? Squishyware?) to run on. Would our souls be compatible with life elsewhere in the universe that might be based around totally different chemistry?

4. Even if they're technically compatible, would our souls even be part of the same system as the souls of aliens? I think that in most belief systems that involve reincarnation, the point of reincarnation is to somehow build upon the souls' prior experiences on earth, to make up for or be punished for things you did wrong, to settle unfinished business, to inch closer to enlightenment with new experiences and knowledge, etc. Can you do that effectively if your soul reincarnates on an entirely different planet? Could, for example, the Dalai Lama, be an effective spiritual leader for the Tibetan people if he reincarnated somewhere near Betelgeuse? And if we are currently operating in 2 different "soul-ar" systems (couldn't resist the pun) what happens if first contact is made? Do our two soul cycles merge into one since our two civilizations would be able to have an impact on one another? Does it happen evenly across the entire human race? Would a member of, for example, the Sentinelese people, who would probably remain unaware of and have no impact on the affairs of aliens, be part of that merged system, or would their souls remain largely in their own bubble?

5. How fast can souls travel? Are they bound by the light speed limit that everything else in the universe seems to be, or can they go faster than that? If they can, does this open up some sort of back-door to FTL travel or communication where all we need to do is off somebody and let them reincarnate across the universe to awaken their latent memories of past lives? Can we encode information on a soul somehow and transmit it that way?

Sort of tangential to parts 2, 3, and 4, the Catholic Church, while not believing in reincarnation, has actually given some thought to the idea of alien souls, and determined that it's possible, even likely, that if there is intelligent life out there somewhere, that they may not be in need of Jesus' salvation in the same way humans are. Since they're not descended from Adam & Eve like they believe all humans are, they may not be burdened with original sin. So there's at least one religion that thinks alien souls may be, in some way, fundamentally different from our own.