Beer mats == coasters
Of course they'd make bad frisbees, didn't think it'd be a question.
Then I read your comment.
Yes, that doesn't make a lot of sense since they go under the steins. But back in the day, you would drink outside and would primarily want to keep dirt and flies out of your stein, rather than the table clean.
So to this day, they're called lids, even though they're coasters (regional variations apply).
Although I've never thought about the rectangularity as being core to the meaning of "mat", esp since there are oval-shaped or semicircle-shaped doormats and such