When flying to the EU from a non-EU country should I use for the booking my EU ID card or passport? Could there be a problem when checking in on the airport with EU ID card only? Example: Flying from Podgorica airport to Vienna airport and checking in only with EU ID card in Podgorica.
The booking is for the airways company. They just want some way to make sure who are you (in order to avoid you reselling the ticket, or that someone else tries to use your ticket without your permission).
They do not care at all which ID you use, as long it serves its purposes.
Passport controls (if any) will be made in a complete separate process.
If a country is involved which does not accept the EU ID card, use your passport.
The airline may not mind but if they need to give API information it has to be a passport, and when they check your passport or ID card, it has to be valid for the border forces of both countries involved (although if there is no exit checks they might not note your passport number going out.)
It is true that the border checks are separate from buying your ticket, but the airline will need to check sometime you have the right paperwork, including visa if needed.