The problem I see with this scenario is: if you're a biological being, planets with a compatible biosphere are probably extremely rare. Just having a biosphere might not be sufficient, if the makeup of the atmosphere is poisonous to you or it's too hot or cold for the way your species evolved. So why would alien civilizations agree to such a thing instead of just adopting a planet and staying there? The whole premise seems to agree that the aliens are all biologically compatible with each other, despite evolving on different worlds. Perhaps they bio-engineer themselves each time they settle on a new world?
> So why would alien civilizations agree to such a thing instead of just adopting a planet and staying there?
I'm guessing you've not read them?
The gimmick of the series is that all sentient species were uplifted from non-sentient animals, and they were uplifted by species that were uplifted, which in turn were uplifted... and none of them knows by whom or by what. The "progenitors", the hypothesized first sentients that uplifted a presumed first generation, are lost to prehistory and nobody knows.
So there is a shared culture and a small degree of shared biology, because each generation of uplifters uses existing models of sentient species to model new sentient species. There are only so many ways to invent the wheel, although Brin was very imaginative indeed in this.
Humans were not uplifted; we're called "wolflings" and as a terrifying novel abomination would have been exterminated upon discovery, were it not for the novels' future setting having us already started work uplifting dolphins, chimpanzees, orang utans and a handful of other species. If humans were not a progenitor species the aliens would have extirpated us -- but we have client species of our own.
He does have answers to most of the objections one might come up with.
Brin's ideas were influenced by Larry Niven, and riffed upon by Terry Pratchett in his early SF works. Later Brin novels riff on ideas from his own earlier ones.
In their time, in the 1980s, I really enjoyed them. I own them all, several in hardback. I haven't re-read them in decades so I don't know how they hold up, but Brin is alive and still active.
In that galactic society, a races' power and prestige are measured by what client races that have "uplifted", taking species that naturally evolved on those fellow planets and then engineering then with intelligence and directing their society. (To some extent in the other direction too: A race might have powerful patrons or grandpatrons that are still around.)
Anyway, point is that for the kinds of races likely to get their manipulators on a fresh planet, direct colonization is often way down their list of priorities. (And perhaps most of their population is already in a Dyson swarm or something.)
Also, the deep history of visits means there's been time for a lot of similar biology to spread around.
it's a bit more involved than that, IIRC (been a while) - the civilisations involved typically migrate around the galaxy, entierly abandoning entire arms of the milky way and moving to another; agreements with other non-compatible civilisations mean that those that remain exist in an extremely hostile environment (the wars between the oxygen and hydrogen breathers, in particular, were notable until they managed to come to this arrangement).
There are also significant numbers of extremely agressive and militaristic civilisations, mostly held back by the laws and customs of setting, who would gleefully seek out and destroy anyone not following along.
