In google classroom, you can swap any saved accounts you have on that device. This is particularly useful if you have more than one google account. See the screenshot below for more details. To access this, please click your initials on the top right of the screen.
As you can see, I am signed in with the main account but below you can see two other accounts with one stating it is signed out. The browser will automatically pick up accounts you log in to. You can now select between multiple accounts by clicking them. It may ask you for your password for that account.
You can add another account to this list by clicking the "Add another account" button, this will let you add an account such as your school account. You can also sign out of ALL of your accounts at once. This can help stop it from shifting accounts while suing classroom.