How to remove shower drain

It looks like you might be able to do this pretty easily.

To remove the stopper, simply unscrew the top of the handle of the drain stopper, then unscrew the underlying screw in the shaft to pull out the drain cover completely. Then, you'll have the cradle-thing that holds the stopper. You might be able to leave that in place (I grew up in a house that had the stoppers taken out but the "cradles" left in place, and two women with long hair would share the bath. We took turns cleaning out the drain) and place the mesh catcher over it. See this video for a demo (It should be region-free.)

You'll probably want to put down putty or some sort of long screw from the drain to the "cradle" in order to keep it in place.

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.