Shower – Can anyone identify what brand this faucet is

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.

It's an installed faucet as far as I can make out. To replace the stems it is not necessary to know the mfr and model (though might be helpful), instead just bring them in to a plumbing store (ideally) or Home Depot.

I once brought my broken version of this part:

in to a plumbing store. Without out my saying a word, the counterhelp simply looked at it, said 'ok', walked back into the stock shelves and came back with the replacement kit.