I'm fairly sure that every other system I've had (most of which, I've built myself) would display the exact model here

This is what I am used to as well, rather than an entire series being listed. I had a quick look at the R7 200 series on Wikipedia and it includes models from utterly useless to ones able to compete with the lower end of the current GeForce 10 series. Very strange.

You could just open the case and read the model directly off the card.