You may apply for an EASA licence for each category of aircraft; but you may only hold one EASA licence for each category. All EASA licences issued to an individual must be from the same EASA Member State, which must be the State that holds the medical records for that individual. A licence holder may change their State of Licence issue, but to do so they must have their medical records transferred to the new State and all of their EASA licences reissued by that State; or apply to the new State for a new initial Medical Certificate (i.e. effectively start again). Part MED allows for a licence holder to obtain a medical certificate from any Authorised Medical Examiner or Aero Medical Centre in any State, but when they do so the record of the outcome will be sent to the State that holds their medical records, which will be their State of Licence issue. (FCL.015 and MED.A.025 refer). As a medical certificate must be obtained before flying solo, the State of Licence issue for an individual will be determined, at least initially, by where that first Medical Certificate is obtained.
Consistent with the above, from 17 September 2012 the CAA will not accept applications for licences or ratings from anyone who does not have their medical records held in the UK.