As other have pointed out, the rules on photography vary from country to country within the bloc. However, the rules governing data protection and the processing of personal data (including photos) come from the GDPR, and very basically say that any processing of personal data requires a valid legal basis.

There is an exception for personal use - the household exemption - but as soon as you cross the line into commercial operations or certain activities such as publishing, creating databases, etc, you lose the benefit of that exception.

That doesn't mean you can't continue, just that you now need a legal basis and need to follow the rules (inform data subjects, allow the right to be forgotten, etc).

So in general, dashcams are fine (unless a local law prohibits them) as you have a legitimate interest in recording your driving in case of an accident. Creating a facial recognition or ANPR database with the same footage would be unlawful, however.