Every game here has a star. Tap it and the game is saved to your list, which you can open from the Saved button at the top of every page.
Saved games live in your own browser, on your own device. There is no account, so nothing is sent anywhere. That also means your list is only on the device you saved it on.
Clearing your browser data or using a private window clears the list too. Some school computers wipe browser data every time you log out.
Every game has its own web address. Bookmark the game page itself and it will stay in your bookmarks however the browser is cleared, and it syncs if your browser is signed in.