Do demons count as animals? There is a famous passage in three of the Gospels (Matthew, Mark, and Luke) wherein Jesus exorcises a man from Gadara. Before he casts them out, he asks them their name and they respond "My name is Legion, for we are many." Jesus then sends the spirits into a herd of swine who rush into the lake and drown.

Source: Mark 5:9

In Rev 13:18 the "mark" of the beast is given as 666... which has come to mean its proper name according to some:

Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.

An interpretation of this number is that it transliterates from Aramaic to Hebrew as Nrn Qsr (Pronounced "Nerō Kaisar")... or the Caesar known as Nero.

So if "The Beast" is a metaphor... no, it is not an animal with a proper name. If you are of a more literal bent... then yes, perhaps the "name" of the Beast of Revalations is 666. (Though the passage says it is the "number of a man and his number is..." [emphasis mine])

Job 40:15 says: "Look at Behemoth, which I made along with you and which feeds on grass like an ox."

That's probably the closest, along with Leviathan, thing in the Bible to animals with a proper name.

There are no mentions of "normal" animals with recorded names.