Who makes this shower faucet handle (logo identification)?
A very helpful support contact at Moen managed to put me in touch with someone at Amati Canada, who suggests that it is an "American Sole" unit. I can't find any information that suggests that's a manufacturer, or a line of products or anything -- so I can't corroborate, but at least I may be on the path to finding out who makes it.

Incidentally, in the meantime, I managed to find a workaround for repairing the part that was broken (I found a small plastic bit that sits firmly in the recess and can be screwed on, acting as a sort of bridge between the screw and the cap, so that when you turn the cap, it turns the handle again). It's not a perfect fix, but it's more than adequate for my needs. If I can get the part replaced, I may still, but it's nice to have options.