Seems most logical and workable that it would be by government. Though I could imagine a parallel structure of species-based councils/institutions existing for matters related solely to a single species, mostly those matters connected to a species' biology - say, age of consent and so on. Sort of like how Belgium has parliaments for both each of it's federal regions AND for each of it's language communities (for matters relating to culture, language, etc.). Not that Belgium is exactly a shining example of a stable and functional federal state...
Do we even know what the definition of a UFP Member is? Is it by planet? By government? By species?
But if representation is for the individual citizens of the federation, and not by planet of birth, planet of residence, or species, then a citizen could vote for any politician running for the council from any species and planet in the federation.
It would be like myself (who lives in Seattle) voting for a representative in the federal government, who is from say florida, because I feel that that person would best represent my personal views, more so that any of the politician running in my home state or district. just as I can vote for anyone for American President, I can vote for anyone to be my senator.
He wouldn't so much represent my state, as my country.