A cat is mentioned in the bible: the lion. So it is incorrect to say that cats are not mentioned in the bible.
The "Lion of the tribe of Judah" is mentioned at New Testament Book of Revelation 5:5. The lion of Judah was a reference to Jesus.
However, at that time the lion was commonplace in Palestine. So it was used as a symbol of strength etc. Of course the lion was gradually extirpated (completely destroyed) from this region and only exists now in parts of Africa and there is a small colony in northeast India.
I don't think it is significant that domestic cats are not mentioned in the bible. The domestic and semi-domestic cat had no significance to the authors of the bible. Other animals are not mentioned either. One author says that the Egyptians were distrusted by the Israelites and the Egyptians had a good relationship with domestic cats so Israelites hated cats. I am not sure about that. The bible was written hundreds of years after the event.