I also work in the EU and privacy on company's machine is a more complex topic that's case by case dependent,but claiming the employer in the EU has no right to access your machine is plain false.

It's true, that the employer can't just enter your machine to look at your personal info just because they feel like it, but they can access it for work related reasons to access work related info, if they have just cause, like a security incident for example.

Ultimately, it's the company's machine, on the company's network, inside the company's premises, on the company's time, so the company can definitely access it for a business use case if they need to, typically with a notification to you to prepare it by removing private info from it or off the screen.

Now if you happen to put your private info there because you're sloppy, and your private info gets vacuumed by accident by IT's mass ingestion of work related data, then you have the right to tell them to delete your private info and they have to comply. That's where the privacy comes in, but it's on you to avoid storing private info on company property, because if it ends up ingested by accident then it's still on you, not your employer if they ingest it by accident.

But I gotta ask, just how hard is it to avoid putting your personal info on company property?