Oh no, there's no way the school system didn't know they were installing monitoring software. It would be fairly reasonable for them to require that in the context of students logging in to access and use school resources (i.e. lessons) and to keep them from just goofing off.

What is not reasonable is the very real possibility that they might have overstepped and made these policies active for all logins on the device, or that they failed to properly disclose what exactly happens when the user uses the domain login. That would then make what's been installed very worthy of the term "spyware" and would probably make both GoGuardian and Google upset with them because it's unnecessarily heavy-handed and pretty much guaranteed to bring bad press.

Part of the reason these things are so locked down is so that management policies can be applied to ONLY the logins that require them without impacting the entire device all the time. If someone tries to sidestep that by grabbing root access and messing with the trusted environment, when the user logs into the managed domain the machine is going to report that its environment has been tampered with and that should pretty immediately bring use of any domain-managed accounts to a halt. You wind up actually jumping through the same hoops (although more discretely and without involving logging/reporting of browser history) when you use your bank's banking app or have Microsoft Teams/Outlook installed on your phone. Both of these things absolutely require their data be kept separate from the rest of the user's data, and very carefully protected from the other apps on the device.