I think there's 2 ways to go about it.
You can either have a literal design, like you said, where the building's shape is like an animal. I don't really encourage doing this unless there's a deeper meaning behind it or you can make the spaces work well.
For example, you might have a multi-story complex inspired by a giraffe. The building might be skinny and tall.
The other way is to have a metaphoric or symbolic approach to it. In this case, the building's form, function or idea is inspired from an animal but does not necessarily look like one.
For example, it might be inspired by a kangaroo. You'd first think, okay, what are the main attributes of a kangaroo?
Kangaroo's hop. They have strong legs. They have a pouch they carry and protect their Joeys in - similar to a vessel.
So you might design a building that is a vessel to carry and protect people in. Perhaps it's form is inspired by this "spring" action of a kangaroo, having a heavy base to represent the strength in a Kangaroo's lower half.
They are just some ideas, but hopefully they open you up to some ideas.
Think about and break down the attributes of a specific animal and begin to think about it metaphorically and literally.
Hope that helps!