> Why is it an issue if Discord thinks you're playing a game?

IIRC if discord thinks you are playing that game all the time, you can't use discord when you play actual games. Or discord may behave differently when a game is running vs. not.

> Does your boss check that you're not playing games on Discord during work time or something?

Oh no, I run discord at my home computer, and use the web interface at work ;)