Должен - да. Надо ли - вопрос :)

Так, похоже и не должен (если конечно права ЕС не получены в обмен на не-ес), вот тут можно почитать https://europa.eu/youreurope/citizens/vehicles/driving-licence/driving-licence-recognition-validity/index_en.htm

Obligatory exchange
When living abroad, you will have to exchange your driving licence for a local one:

if your licence is lost, stolen or damaged
after 2 years of usual residence, if you have a driving licence valid for life (only if required by the country where you live)
if you commit a traffic offence in the country where you live