All right you pervo wags, yes, it's funny that Jon enthusiastically shows a woman his huge wide spongy tongue. That's 1/3 of the joke. Another part is that Jon is hanging on Liz's every word to the extent that he obeys commands intended for a dumb animal. Everyone gets the gag.
The bonus-level for the thinking Garfield fan is that Jon is so desperate for approval that he seems willing to follow any directive Liz would give... So surely his vet has told the man not to feed pasta and melted cheese to his 27-year-old cat. It's Liz's one demand that Jon would never follow. Because Jon puts more value in Garfield's loyalty than his chances with Liz. Because he's weird and screwed-up.
Cutest things: Good kitty Garfield behaving himself as Liz examines his eye. Also Liz's smile at Garfield's cooperation in panel 2.
Liz's office is suddenly barren as Jon's breakfast nook. Before the nitpickers cry foul, I remind you this is a prime example of flexible comics reality. Garfield is about stripping everything to the bare minimum elements of visuals. And humor. And life.