The Cat Plush is the equivalent of Wilson in the movie Cast Away. A simple stuffed animal that You can talk to. No other obvious implications have been discovered thus far.
Sleepy Cat has its own comic and it can be inferred that Sleepy Cat is a pop-culture icon in the "Lone Survivor" universe akin to "Hello Kitty".
Attempting to give Hank the Cat Plush will cheer him up. Hank will refuse to take it because he knows you're more attached to it and give you a Burger Bun as gratitude. (This only works if you have given him all five Health Tonics and if you are on New Game+.)