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You still need to ask permission for storing identification data, regardless whether user's browser is told to save cookies or not.

Yep. The legislation has nothing to do with cookies, it’s about storage of unique identifiers. The actual method of storage is largely irrelevant.

For GDPR yes, but there is also a separate and older EU regulation that is specifically about cookies (among other things).

Yep, I mentioned PECR in another comment here