Sorry I haven't guessed apart from the throwaway first guess. :P I'm gonna guess the Invincible, although I don't expect it to be correct...
Twice Your Protector,
Three Times A Foe,
Each Time Met Royalty,
And Would Not Let You Go
It is three times a "foe" by my count: The attacks on Madain Sari, Brahne's Fleet, and Alexandria. It's a "protector" when you fly it into Memoria, and also when you get the airship Invincible in FFIII.
Each time met royalty: Queen Brahne is present at the second attack; Princess/Queen Garnet is present at all three, although she wasn't a princess at the time of the first. The protector part is even more sketchy, you do "meet" Cid when you fly into Memoria, but I haven't gotten that far in III and I can't find anything about meeting royalty in that context.
"And would not let you go" I can't really see how that fits. The same goes for the whole second stanza actually xD But yeah, I don't actually expect it to be correct.