How does one identify compatible shower drain covers?
Drain covers are specific to the tile drain manufacturer and can be difficult to find. Yours, however, looks to be of a common type and is very much like the middle one in this picture. I would take careful measurements and make a paper and crayon "rubbing" of your drain frame and go to the best plumbing supply shop in town (try a really good tile supply shop as well).

This is a website you might try: https://www.plumbingsupply.com/draincovers.html



It is odd that there are no screw holes in the drain scupper, the cut-outs in the tile frame are made to provide screw clearance. There is a chance that your tile guy tiled in a drain frame that was not the style meant for the actual drain assembly. If that is the case you may have to resort to a "snap-in" style retrofit cover. They never look as nice as the original but it's better than an open hole.