Most of the characters in FFIX *did* have last names:
Zidane Tribal
Vivi Orunitia
Adelbert Steiner
Freya Crescent
Eiko Carol
Quina Quen
Amarant Coral

If I'm not mistaken in most of the FF games characters had first and second names (maybe not the oldest ones), until FFX when for some reason they didn't.

I don't know about Garnet's real last name though.