Additionally,
1)I can say
the animals' names. <noun>
Meaning: called by; e.g., the elephant is called Sophie, the bear is called Max, the lion is called Miao, and so on.
2)I can say
animal names. <adjective> Cf. animal sounds
Meaning: called by; names of animals (ambiguous)
3) I can say
the names of the animals.
Meaning: nomenclature; e.g., elephant, bear, lion, etc.
Note, adjectives don't take determiners (a/the) or possessive
's.
Also consider,
Pat: What's
your dog's name?
Sam: His name is Rufus.
Pat: What's the
name of your dog? (as in, what type/breed or name)
Sam: He's a Shepard.
Sam: He's called Rufus.