What do you call this button-shaped thing?
Solution 1:
It is called a rivet fastener. If it is plastic, you can call it a plastic rivet fastener. If it is metal, you can call it a metal rivet fastener.
Solution 2:
A blog post titled Peugeot 206 Rear Light Replacement shows the actual image you posted in the question. In it they call it a plastic bung. But rivet fastener seems like the better name here since bung may not be understood by many.