In the phrase "No more than two cats or dogs", how many total pets are possible?

The phrase is ambiguous. The fact that you are asking this question in the first place shows that it is a poorly worded phrase.

I don't see any ambiguity. By the use of the word or, the phrase means you may have up to two cats or two dogs. Because it sidesteps the question of having one of each, I would guess that the writer of the contract didn't consider such a possibility. Neither does it seem the writer thought to exclude monkeys, elephants or any other weird pet choices.

A tighter wording might have been "No more than two pets (limited to dogs, cats or one of each)."