Chromebooks must be enrolled under Chrome Education in order to be recognised as 'school property' and therefore become controlled and monitored by schools. Signing in with a school account will not enable these extra features.

However, signing into Chrome on any computer (Chromebook or not) will send search results into the Google Account's search history, which, if the Google Account is a school one, is monitorable by IT to see search history and browsing history etc.