I think it's an option at the bottom of the "inputs" section in the settings. It's one of the "control other devices over HDMI" settings. You should be able to turn it off so other devices won't change the TV settings automatically.

Yeah search for "low latency mode" in input settings. That's what game mode is.

Just found this answer 2 years after you posted it- I spent all morning trying to figure this out and your two year old answer helped- ThanK YoU!!

deu certo, parou de ficar aparecendo a mensagem modo jogo ativado, agora posso deixa sempre ativado ou desligar

I'm getting the same annoying message when I full screen while Extending my Android's screen to my Hisense TV. If I don't put it in FS it won't cause this to pop up. Since, I have 3 different elements I didn't know where to look. But, your issue simplifies it some. So there should be some setting in the Hisense tv causing this issue.

In this morning I get the same massage, it something annoying. After last update My Ps5 .I got this problem with My new TCL TV . And until now I can't figure out how to turn off the game mode.

Thank you! Late comer here!

Check that inputs settings for auto low latency mode