I don't know about that specific browser. But there is something similar in use in schools here. Basically they know when you're leaving the browser window ( e.g.: to open a text file or a different browser ). Other features probably are that they can monitor you through your webcam and/or listen to your audio as you are making the test. The reason is so you don't have somebody under your desk reading you the answers. Or you don't have your course book open while making the exam.
The school could honestly provide the alternative to go make the tests in the school itself under supervision on paper or on a school device.
It's a secure anti-cheating browser, not a secure privacy/anonimity/hardened browser.
It's not meant for day-to-day use. Only for making the exams.
You can always get a secondary cheapo device purely for the exams ( or an old laptop).
How do you expect a degree to be worth something if the only proof they have of you taking the exam fairly is "trust me bro".
If you take an exam for AWS in a managed center you do it on their computer with a person watching you on a camera and listening to your desk with microphones.
If you take the same exam at home you're monitored by a remote employee constantly looking at you. You have to show the room you're in to prove there's nobody there.
If you don't want the latter you take the former.
From my personal point of view I feel the school should offer a way to do the exam in person and on a device provided by the school which they know is secure if you so desire as a student. Perhaps that's not possible in the college you're attending. Or it's a fully remote course.
In any case. They school needs some guarentee that you aren't cheating. Either by attending in person, or by having a "secure browser".
Secure browser meaning a way for them to check you aren't just searching the web/LLM/course material for the answer ( or an external party helping/doing the exam for you ).
Unfortunately none of that is possible without looking at your room, listening to the environment you're working in and have you work in a browser they know isn't tampered with and isn't being minimized/left unfocused to do something else.
If you're worried about what happens with the recording post exam you're better off asking them rather than assuming they use it for whatever. They likely have to store it for a certain amount of time before it's deleted.