More often, overflows screw into a ring behind the back of the sink, and you can unscrew them. Rubber gloves will give a better grip. You may find it very hard to reach behind the sink to hold the outer part in place so you can screw a new one on. Ceramic sinks often don't have that, there is a channel moulded into the porcelain, which joins up under the plughole, with no visible pipe.
A photo of your actual one would help. The one in the link looks like the plastic fins have a barb and will break if pulled out. It might be just an ornament of no practical function.
Rubber gloves will give a better grip. Put a finger in the hole and see if you can twist it. If not you can pull, if you don't mind that it may break.