Is there such a a thing as a single European pilot's license?
Is there ... a single license that allows a pilot to operate any European registered aircraft in any European country,


Yes.

According to the UK CAA
an EASA Part-FCL licence can be used to fly any EASA aircraft in any EASA member country.

I have interpreted your "Europe" as "EASA member" in order to avoid ambiguity about Switzerland, Croatia, Kalingrad, etc.


  Most aircraft in Europe are EASA regardless of state of manufacture or registration.





  The UK currently grants several levels of licence:  
  
  
  UK National Private Pilot Licence (NPPL) 
  UK Private Pilot Licence (PPL) 
  EASA Part-FCL Light Aircraft Pilot Licence (LAPL) 
  EASA Part-FCL Private Pilot Licence (PPL) 
  EASA Part-FCL Commercial Pilot Licence (CPL) 
  EASA Part-FCL Airline Transport Pilot Licence (ATPL)