It looks like its grouted in. Take measurements of the outside dimensions of the square and the center to center dimensions of the screw holes to find a replacement.
To get it off, lightly tap it with a dead blow hammer, rubber mallet or a steel hammer covered in a couple layer of towel. You want to lightly fracture the bond between the strainer and the grout. You likely will loose some grout when its removed, but its cosmetic and learning to repair that will be one of your new things to learn about DIY. You could also preemptively remove the grout around the strainer with a tool made for that. basicly a knife that you pull a carbide tooth and it scratches the grout away. Could be done with a dremel tool too if you're very careful not to nick the tile.
You can get a 18" length of wire.. copper, bailing wire.. whatever you can put a bend on that you can insert into the grate and then basically hook the strainer. You woould bend the end of the wire about 2" from the end to a point. Insert the point into the strainer then lift it so that the short end comes up a second opening. You now have a hook. then twist the hook around the long length of the wire and you can now apply force upwards on the strainer as you tap with the hammer.
