You're supposed to push down on the hinged thumb piece until it clears the plastic tab in the base and then slide the top of it toward the cooler to release the clip. Or it may be easier to use a small screwdriver to pry the lower part of the tab outwards away from the cooler while you are pressing down. Then you can sort of wiggle the whole cooler around to unclip the other side.
There are plenty of clip-on coolers like this and the plastic base may screw-in to a backplate or attach via push-pins. In any case you can get to that much more easily once the cooler part is off
That one has thumb clips on both sides so you can undo both before trying to unstick the cooler from the CPU and thermal paste by twisting. The angle on those clips though tends to force it tighter toward the tab when you press directly downward, which is why you are having problems.