Your choice would be "мой ученик был котом (/кошкой)" or "моя ученица была котом (/кошкой)".
Me too! Why did no one answer you?
(I'm guessing it's because of the У, and the latter might just be without it? Don't take my word for it though.)
The "У" here does denote possession. Literally, the sentence means "Within my student ('s possesion) there was a cat."
But! As Vlad illustrated, you can't simply drop the "У", because half the sentence is in the gentitive form due to its presence. You would have to reform all the nouns to be in the nomative form.
After у we use Genetive. If we were speaking about students, we would say "у моих учениц". Ученицы is either Genetive Singular and Plural Nominative.