Lyric discussion by graviton 

Cover art for Thunderbird lyrics by They Might Be Giants

Yes, Thundebird used to be a cheap type of fortified wine, that was notorious for being what winos and bums would buy to get drunk on little cash. No idea if it's still around.

The "till T-Bird takes her dad away" thus can possible have a darker meaning, beyond the play on the Beach Boys song. The singer is an alcoholic and will soon die if he doesn't give up his drinking, leaving behind a daughter.