Having looked into it a little bit (i.e., having read the wikipedia page about acoustic fingerprinting) I see that there are quite a few implementations in existence and that they appear to be fairly reliable at identifying songs based on the fingerprints -- which is really cool, first of all. Taking that for granted, it seems like facebook could solve its problem by creating a comprehensive database of songs which its music content partners offer.
In particular, I'm thinking of a database with entries containing an audio fingerprint, data about the artist, song, album, and so on, and links to the song on each of the partners' services. Then it would be possible to initially loop through each song offered by the partners, generate an acoustic fingerprint, check if it exists in the database[1], and then add a new record or add a link to an existing record as appropriate. This would be a huge process, of course, but afterwards you'd only have to perform this step for new songs, and you'd ultimately end up with a very impressive database of music IDs.
Anyway, that's undoubtedly a simplistic idea of what could be done and I'm surely glossing over some major obstacles, but I enjoyed indulging in some wishful thinking about it all the same.
[1] As I mentioned, I don't know anything about acoustic fingerprinting, but intuitively I can imagine it's probably not as easy checking if that fingerprint "exists" in the database. More likely, you'd have to check how similar it is to existing fingerprint and choose some threshold above which it's sufficiently likely to be the same song.