all the hardware programs i can think of list only the amd/nvidia chip model not the brand of the board.
can't really think of a way to find out without opening the case. perhaps you can scan the programs installed and see if there is any brand specific programs installed. generally they come pre-installed so you may get lucky and see a handful of things from msi/asus/or other brand specific stuff.
edit: rechecked gpu-z and though it only lists the amd chip i have if i clicked the "look-up" button next to it, then it took me to a page with my actual card on it. specs and all included. so maybe, you can get that running on the pc and see if ti does the same for you