How should this be opened: prying it off or unscrewing it?
This words on the cover are: "florestone madera ca"
The lettering on the drain cover leads to a number of similar (but not exactly the same) products all over the 'net. What is consistent about this particular company's line of products is that the words "snap-in" appear frequently.
The image above differs from yours in that your "tab" area is much thinner and does not show clearly any tabs. If you are able to shine light into the area of the slots and are able to see tabs, one can expect that it too is a snap-in cover.
Even a rather comprehensive installation document for plumbing a Florestone shower pan includes nothing about the drain cover. This is consistent with one being able to see the removed cover and recognizing that it is a snap-in type. In place, it's not so easy to determine that aspect.
Image courtesy of Decour Glamour web page.
If it says "Florestone Madera CA" just put something in the holes, pull, and it pops right up. If you want, you can put two dowels or screw drivers with round shanks in the holes and put another long screw driver parallel with the floor across them, as leverage, and turn the drain to break it loose. Just has hair in drain and slows the water draining out.
I have a cover EXACTLY like yours. Mine just lifted right up. Not sure if there was supposed to be some kind of sealant, but it would not surprise me that the installers "forgot".
Hi It seems like the oblong holes indicates that a special tool is needed to unscrew it, so I would attempt to unscrew it by improvising a tool in those oblong slots. Frank