Slang for girlfriend, usually a secret girlfriend. Cat relates to a female girlfriend due to her anatomy, a pussy ... my pussy cat, They drop the pussy, and it becomes just "my CAT", or cat.
When I go home on leave, I don't know which to go see first my wife or "my cat."
The person is trying to keep his girlfriend secret by talking about her in code, in case someone else hears them. He can plausibly deny that he was talking about a kitty cat, not his secret girlfriend that he has on the side or down low.