How to remove shower drain cover
I have one very similar to that.  It lifts out vertically.

There are six keyed slots in the drain tube that the cover slots into.  So you have to lift it keeping it level.  Place something under two sides of the cover and lift together, if you only lift from one side it will twist and won't come out.

There will then be a plastic hair trap guard that just lifts out easily that you can clean.