The real question isn't whether or not GoGuardian is installed--because the Chromebook is going to be subject to the school's policies while the student is logged in, assumedly doing classwork. The real question is whether or not it's being disclosed to you when you initially login to the school's domain that this software is being installed.
Installing something you were given a chance to read the EULA and disclosures for is fairly reasonable, because you would have the opportunity to decline (and then they probably wouldn't be able to login to the schools network). The school using MDM to install the monitoring software _without disclosing this to the user_ would likely fall afoul of the law, because yeah if you paid for it you are the owner and what you say goes. Third parties are not allowed to install software on equipment you own without explicit permission. Additionally, to be even remotely ethical, this needs to be disclosed clearly and the first time the school login is used and the software installation is about to occur--not buried inside a school handbook somewhere.
If the school has been so bold as to ignore the need to disclose that they're installing something that records and reports all web browsing activity (which GoGuardian is designed to do), or if this software applies to other logins on the system which will be used by people who did not consent to being monitored, then the school needs to start hiring _competent_ people to administrate the network, because doing this with minors involved is just really begging for a judge to slap the school system hard.
The school system can (and should) make that policy only apply when the kid's school login is being used--so if they try to fob you off with that silly excuse, feel free to go ahead and start talking to lawyers. Google would not be amused to find some rinkydinky school administrators making claims that the attested environment used by ChromeOS may be trivially compromised by other users and thus justify installing the monitoring software to be in play all the time. Rather a lot of work has gone into creating that environment.