That's one of those aspects I want to fix, the scale of the UFP's territory sounds big, but it's about as big to the relative Milky Way Galaxy as a single city block in Singapore is to the the scale of the rest of Earth.Perhaps there is a UN-like structure in place, but the Federation is not enough of a serious galactic power yet to be invited to that table. (A major pet peeve of mine is how Trek often underrepresents the scale of the galaxy and the sheer amount of exploration to be done.)
It's really a drop in the bucket.
Also, finding a UN-like structure that far apart is pretty hard when communication distances / styles are so vast and different.
It's hard enough to communicate over the existing distances that we have on Earth, imagine how much harder it is when communications have to travel over MUCH further distances along people who might not have many things in common linguistically.
One of those things I really appreciate about DISCO's recent last season finale was how alien / difficult it should be to communicate with some new species you met who you have no common background.