The guestbook and puzzles don't need an account. So how do they know which notes are yours?
Your browser remembers. When you add an entry, the site saves a small ownership marker (sometimes called a "token") inside your browser. Next time you visit from the same browser, it sees the marker and shows edit and delete buttons on your entries.
What you need to know
Use the same browser on the same device to keep editing.
If you clear your browsing data, you'll lose the marker — the entries stay, but the edit buttons go away.
Private/incognito windows don't remember between sessions.
Switching phones or computers means you can't edit past entries from the new device.
Guestbook guide
How to use the guestbook