"I have many aunts."
Translation:Mám mnoho tet.
it is Genitive, which happens to be TET. Many, a lot, little ... behave as numbers from 5 up.
I thought in other "have" constructions, eg. Matej has a young wife, the wife has only been in accusative form, not genitive. (But I may be wrong about that, of course.) Is it purely the addition of mnoho which requires the genitive form of aunts? If I were to say "I have young aunts" - would they be in the accusative?
You're correct. "Mnoho" requires genitive, while you would use Acc in "I have young aunts."