@Clementine ChromeOS has a Linux layer. If you own this Chromebook, look up how to enable it and install Firefox. If you don't, ask your administrator to do so, or just get a normal netbook.
This isn't just about ads. If any website can request a copy of your entire personal profile, they can 100% attach it to your real name and whatever information they already have on you, and use it against you.
They could use it to dox you or blackmail you, they could use it to generate more convincing malware based on what has or hasn't worked before (think about those fake download buttons, QR codes, and messenger windows), and that's just off the top of my head.
We don't know what kind of data they're actually collecting, so who knows what it could also be capable of.