They're handing a car to someone, and they need to verify who that person is - for liability purposes, etc.
In many European countries, a driver's license is not considered to be a valid identity document, it's explicitly legislated that identity documents are either a passport or a national ID card, and nothing else. So it makes all sense that all the company procedures state that they must request proper ID (and verify it in online registries of stolen IDs where possible), and strictly ordered to not accept any "not-a-document" substitutes as any gaps in policy would be systematically abused by fraudsters.
This is not specific to car rentals - pretty much every institution that needs to check ID will not accept drivers licenses. I don't know if Netherlands has it the same, but if my local store would want to verify my age for buying alcohol, a drivers license would not be acceptable for that - they're mandated to verify an identity document, and drivers license is legally not an identity document. However, it's worth noting that this is a relatively recent change, 5-10 years ago the penetration of ID cards was much lower and drivers licenses were accepted much more widely, but they are not anymore. This may explain some mismatch in expectations, as one institution may be unaware that another company has changed their policies.
It also seems plausible that they might extend the same policy to non-European customers for simplicity, and also for real risks - for example, it's nontrivial to verify the validity of all the many forms that a USA drivers license might take, documenting how you should verify USA passports seems simpler.