What if: Han calling father-in-law Anakin 'Pops' or something like that in order to be sort of cheeky and pretend he isn't the scariest guy in the galaxy
This is absolutely what happens in the double agent AU.
The thing is, Han was actually not that scared of Vader. He shot the guy down that one time, right? And anyway he doesn’t believe in all this mystical mumbo jumbo, and that mask is honestly kind of tacky in his opinion: base intimidation tactics, not even very clever. He’s seen better.
But then he meets Ekkreth, also known as Anakin Skywalker, hero of the Clone Wars, double agent extraordinaire who regularly lies to the Emperor, whose not-really-that-intimidating mask is apparently a life support system, who has literally killed people for having insufficient faith in that “mumbo jumbo,” who’s a damn good mechanic and a better pilot than Han would like to admit, and who is also, by the way, Leia’s father….that guy is terrifying.
Anakin is kind of amused by it, tbh, because he actually really likes Han. He keeps trying to turn their past encounters into in jokes, but it takes Han a while to figure out that’s what he’s doing, rather than, say, trying to threaten or goad Han. It’s all, “Let’s see you try to shoot me down this time, Solo” and “This mission requires a good bit of acting - better leave Solo behind” and also probably a lot of mercenary jokes and of course constant teasing about Han’s “lack of faith.” (Though tbh, now that he’s hanging around the three Skywalkers pretty much all the time, Han is finding it harder and harder to hang on to his lack of belief in the Force.)
I imagine Han probably responds by giving Anakin all sorts of insulting nicknames. Like, at first he probably just calls him “Old Man,” all the time, but that becomes really awkward when it goes public just how not old Anakin actually is (and when Han realizes that there’s effectively the same age difference between him and Anakin as between him and Leia, which is…definitely awkward).
So then he switches to calling him things like “Pops” and “Rust Bucket” (something that initially horrified Leia, but is actually probably Anakin’s fave nickname). Anakin in turn refers to Han as “Kid” and “Nerfherder” (which always makes Leia cringe in embarrassment).