If you play on https://ducklings.io/ directly this should't be an issue, but it seems like you're playing on itch.io, which does have this issue because it embeds games just like other game portals. If you're using Chrome and you see this icon in the top right: 
third party cookies are disabled and so the progress won't save. By default Chrome only has this happening in incognito, so if you haven't changed your preferences this shouldn't be an issue (until they change it in 2022).
On Firefox third party cookies are always disabled by default I believe. So it only saves when you play on https://ducklings.io/ directly.
On Safari things are a bit weird, I think safari tends to delete stuff after a while whether you want it or not.