This is a little tricky because the question and answer don't exactly match up... The correct way to ask is just as you stated "What is your favorite animal" but the expected and "correct" answer is in plural, usually answered with just the name of the animal in its plural form such as "dogs." Never thought of this before, kinda weird!

The first one is only valid if the question is 'What's your favourite animal for eating?' because you can use 'dog' as a shorthand for 'dog meat'. I'm guessing that's not what's being asked. The second is correct. However, you would usually say 'Dogs are my favourite animal.' Another possibility is 'The dog is my favourite animal', but that would also be uncommon. If you need to talk about the generalized idea of all dogs, then you say 'the dog', or 'dogs'.

I have no sort of clue what my favourite animal is but my favourite colour is blue

The correct answer would be neither. "My favorite animals are Dogs", works much better.

cat