My favorite animal is a dog or a cat. My favorite animals are dogs and cats. Since you have multiple favorites, the second sentence is correct. If your favorite animal is either a dog or a cat, the first sentence would be correct.

What is your favorite animal that you like to have as a pet? What is your favorite animal and why?

Just to add to Matt’s correct answer: Animals are countable (almost always), so you need a determiner in the singular. The determiner is usually an article (a, the), a demonstrative pronoun (this, that, these, those), some, any, no, or a possessive (my, your, our, etc.). Omitting the determiner in the singular would be wrong. Meat, on the other hand is (usually) uncountable, so if people say “I like cat”, the meaning is that they like cat meat — to eat. In context, you’d probably be understood, but it’s a very common point of English grammar, so I’d suggest that you work on mastering it. Edit: Corrected “relative pronoun” to “demonstrative pronoun”.

'My favorite animal is a dog' is wrong, it means one dog, like 'my favorite Disney character is a dog'.. "What's your favorite (Kind of) animal?". 'dogs'

Thank you!