Suppose you win a game in a noncompetitive way, when you know your opponent cannot feel good about his play. Maybe he dropped his queen early, to a simple tactic that should have been evident for a player of his level, for example.
Do you say "Good game" when you shake hands? Or do you say nothing at all?
Why not just:
Thanks for the game.
You wanted to play a game, and your opponent gave you a game. Played well or not, you could thank him for giving you the opportunity to play.
In short: Yes. It is a polite way to acknowledge an enjoyable game.
"Good game" isn't the same thing as saying, "well-played game".
Saying "good game" just means you enjoyed playing it. You could have enjoyed it for many reasons. For example, it could mean something as simple as, "thank you for being polite and not stalling or offering draws from badly losing positions", or "not being a d*** when you lost".
It could even mean, "yes, I clearly won, but you kept me on my toes".
"Good game" does not imply that the opponent played well in any way, and is the normal polite way of thanking a person for their time. If the person is rude to you, or obnoxious, then you wouldn't probably feel the desire to say "good game", no matter how well they played.
In general you say something like "Good game until ... That was a shame".
Maybe he dropped his queen early, to a simple tactic that should have been evident for a player of his level, for example
So, in that case "Good game until you dropped your queen. That was a shame because it was a real fight until then". It may be a small white lie but it is less transparent than the standard "Good game" which comes across as formulaic and lacking in sincerity.
In Go (which I discovered after Chess, and ... much prefer now, I have to confess), we usually say "Have a good game!" before we start (the meaning being: let's both play a nice game, and enjoy it), and "Thank you for the game" at the end, said by both parties. There can be tons of variants, but there is usually a good attitude from both players before and after the game. If the game was well played on both parts, it's often pointed out by saying something like "This was a really nice game!", as we are both happy we played it (even when one loses, if the game was well played by both, it is a good feeling to have lived it and contributed to it). And often the game is then discussed by replaying key moves of the game and exploring what each could have done better instead of those moves. Being fiercly fighting on the board during the game shouldn't lead you to be rough to your opponent outside of it, so saying "good game" to your opponent for a game where the opponent blundered seems a bit misplaced, even hurtful. Just saying "Thank you for the game" is enough (and hopefully doesn't convey the same feeling. Intonation is key.).
I try to, but in the case of egregious errors like that, I'll typically slide that into "...at least up until move 15" (or wherever the queen fell). If it's not just dropping the queen to a pawn, like maybe it was a long bishop move for example, I might add something like "Long moves are sometimes hard to see."
We all have those kind of games and we all feel horrible after it. I try to at least say something that might relieve the stress a bit. maybe ask a question about a moment that occurred earlier in the game, if I can come up with one.
To me the important thing is to be encouraging. Sometimes that can be really difficult.
Whatever you choose to say to your opponent, consider also saying something along the lines of:
Perhaps we should have a rematch sometime?
Which is a polite way of acknowledging that they messed up and offering them a chance to redeem themselves without actually embarassing them by stating that you know they played badly.
Not directly pertinent to your question, but I'd like to add that if you do end up having a rematch with them and they make the same mistake again then it may be a good idea to that they in fact aren't aware that they made a mistake and instead inform them what their mistake is and why it's a bad move/strategy.
