The smart winner would not agree to doubling up.
Casinos don't let you perpetually double up, because eventually you will win and even things up, negating all the previous wins for the house. Then you play till you're ahead and then quit on them.
In pool, it also screams hustle. Sometimes it can just be the desperate last efforts of a loser to make up ground or get out of the hole, or it could be a trap. So many variables. If the sets were real close, then the double up is just the loser trying to catch up. If the sets were beat downs and lopsided....then that is suspicious because why would the loser feel so bold and confident to ask for double? Either way, a masterful hustler can keep sets close and dump. So that isn't even a valid way to look at it.
In this thread, the loser quit which is proper protocol. The winner did not quit on him, even though that is still the winner's right (although it is dishonorable and nitty to do that). The loser quit because of the refusal to double the bet. The winner did nothing wrong, nitty or crappy by refusing a double against someone he's ahead on. If he was willing to continue at the same bet, he's doing the right thing.
What does this say about the loser's motivation? Hard to tell. If he's a hustler he may not want to waste his time winning back the sets at the same bet. Because that will surely alarm the early winner and he'll just quit once even.
Lot of speculating, have to be there in person to see the caliber of player, see what is happening.
I see no issue with anything that happened.