Is "Good Game" a impolite thing to say in chat after winning a game?

I mean it is a nice thing to say regardless of how the game went. A game of chess against another human is always nice in my book. I just like to say it if I played someone a couple of games. But once in a while people react as if I made a sarcastic taunt if I won. 

It is a nice thing. As you said, it is surprising how many people take it or hear it sarcastically.

It is sad that so many people seem to be wound so tightly.

Perhaps that phrase has been overused and saying the same thing in a different manner might work.

Some people will get angry if you say gg, some people will get angry if you don't say gg.

It is a polite thing to say.

I believe it is the responsibility of the loser to show sportsmanship and say "Good game". However, if you are someone who often says good game after losing, then it justifies you saying good game after a win.

It's not impolite to say good game. It's just impossible to please everyone.

Maybe I'd avoid to say it when you just delivered a scholar's mate.

That would probably be a good idea.

As once Magnus said, "Who the hell cares!"

I always say, 'Thank you for playing', which is more neutral perhaps.