Thank you for your answer.
https://johannpoufpouf.itch.io is authorized for cookies (Exception handling).
"Always use private browsing mode" was ticked. I disabled it, but this did not change anything.
Enhanced Tracking Protection is set to "strict". For now, I am not ready to set it to less strict than it is now. Maybe this is the issue.
It won't stop me from playing, even if the progress isn't saved.
Have a nice day!