How does one identify the motherboard in the Apple System Report?
When you select Apple, About This mac, the window that pops up will give that information already (10.8'ish and newer)
Serial Number (number here)
if it says "serialnumberser" or something odd after "Serial Number", then it did not get set when the board was replaced. That will interfere with quite a few things, including signing into iCloud and using the app store. You'll have to take it back to the shop to have them finish what they started.
If you go into system report though, select the category header "Hardware" if it's not selected by default. In the list to the right should be a line starting with "Serial Number (system):" which will also display the computer's serial number. (it also may say something defaultish like "serialnumberser" if it's not set)