I think vielen is used for the dative case. Like Ich danke vielen Leute. Whereas viele is used for the accusative (like in this example) and nominative case eg Viele Leute trinken Milch.

But "multiple" is not the same as "a lot". Two animals is multiple animals, but it isn't a lot of animals.

Because that would translate into "Do a lot of animals have you?"

I agree and that's why I commented to duolingo. They are splitting hairs. Mehrere = multiple, viele = a lot, eine Menge would have to be accepted as well. I can split hairs too.

Wie viel? = How much? Und Wie viele? = How many?

Menge is "amount".

But eine Menge before another noun idiomatically means "a lot of ..." rather than merely "a (certain) amount of ...".

Because Tieren with its final -n is in the dative case, but in this sentence, you need the accusative case: Tiere. (The same as the nominative case in the plural.)

Have they got a lot of animals? = Haben sie viele Tiere?

The subject pronoun would be lowercase sie in this case rather than uppercase Sie. Everything else would be the same.

vielen Tieren would be dative plural, but "a lot of animals" is the direct object of *to have" and so needs to be in the accusative case.

Accusative plural, if there is no article, is viele Tiere.