Pictures?

Here are some pictures

There is no visible screw holding it down, which means that it may snap on to something in the assembly you can't see. However, I think the main problem is that it is basically cemented in place by the grout on all four sides. If it were me, I'd use the grout saw to go down about 1/4" all around, and the try to pull it out again. This will be difficult, especially clearing out the corners and dealing with the side where the tile butts right up to the drain. In my experience, even a small amount of residual grout could hold that drain in place. If you can't get it loose around the edges, you might have to sacrifice it by using a grinder to cut a square hole in it, leaving just a narrow perimeter of metal which can be pried out.

The only other thing I can think of is to see if you can at least cut into the grout along the "bottom" side of the drain in the photo where there appears to be a tile piece missing. If you can get the bottom edge of the drain exposed, you might be able to get a small pry bar or the tip of a screwdriver under the edge and then gently lever it up enough to break it free of the grout. If you try this, make sure to cushion the tool against something thin, like a ruler, so you aren't exerting leveraging force directly against the tile and breaking it.

I wish I could be more encouraging, but this is a difficult task given the poor manner in which it was installed. I hope you have a decent level of skill with tools as this is not going to be easy, unfortunately. Good luck.