EU citizens can bank relatively easily with banks in another EU countries.
1) You need to be born in the EU; (otherwise it might be more difficult)
2) You need to consider the languages offered by the bank; (local and sometimes English)
3) You need to know if they specialize in a certain 'non-resident' area; (cross border traffic, expats)
4) You need to have a good cover story; (I want a euro/zloty account, as I intent to {move,buy} X)
Online banks are easy;
bunq, n26.
Retail banks that do online opening: DKB.de, ComDirect.de, HSBC.fr.
A lot of banks will open in person, make sure you adapt #4 to fit the bank.
It is not forbidden to open a account elsewhere in the EU, tax information will be exchanged!
Banks use verification methods that defeat 'the
darks', so only open with real ID's.