Saving game progress without creating accounts for children

I am looking at options for saving progress in short browser games without asking children to register. The Dad4Kids page I have in mind is Reading a Clock — Free Telling Time Games for Kids. This is an architecture question, and I do not have a deployed Lambda setup to compare yet.

For a first version I would keep progress in the browser. If a parent later wants to move it to another device, would you use a short-lived export code backed by API Gateway, Lambda and DynamoDB, or put the saved progress in a signed token?

The saved data would be the exercise identifier and completed level, without names or dates of birth. I am unsure how to handle repeated imports and expired codes without adding a permanent user identifier. I would appreciate examples of how others have handled that part.