In The Netherlands, employees enjoy a reasonable level (much stronger than this scenario) of privacy even on company equipment [1]. That is: the company cannot randomly access an employee's mailbox. Of course, in case of employer-employee dispute, things become hairy - but even then, the company is not allowed to go on a fishing expedition. Though they can go on a confirmation run, looking for specific evidence, e.g. mail from Glassdoor.
Don't know about other EU countries, but at least NL deviates significantly from the sketched scenario.
[1] https://blog.iusmentis.com/2017/11/22/wanneer-mogen-mailbox-...
In most EU countries, tracking company-owned hardware is explicitly okay, and where it's not mentioned in law there are judicates that make it OK.
As I said, even the government and its wholly/partially owned enterprises are doing it, and working as a contractor for the government here requires you to track usage of your employees' workplace computers too, so I can't see how it could be in any way illegal. Same thing with working for banks and insurance, and I bet there are more cases.