I think the emote was actually originally named after the literal 90s game pogs, at least that that's what the old writeups suggest. The gootecks guy had recently done and won a goof video about playing pogs with some other streamers when the emote was first introduced, so it was named pog champ after that happening.

The play of the game thing also makes a kind of sense as a post-hoc rationale though, since the emote was used for the exact kind of surprise/excitement that you'd expect after seeing some crazy play being pulled off.

That's a backronym. The original POGchamp emote referred to two separate events. The picture of Gootecks making a surprised face was from an outtake video of him and his co-host, when a cameraman bumped their tripod and shook the camera. The actual POGChamp term comes from a slightly later, scripted, video where they were playing POGs and Gootecks trumped his opponent by using a MadCatz fightstick as a slammer. At the end of the video he makes a smug face and says "pogchamp". The "Play/Player of Game" thing came around later when people tried to put a different meaning to it.

I think you were supposed to use them like Tiddlywinks, but all I remember the kids doing with them, was collecting and trading the things at school.