I think trying to frame something you give away to everyone, always, without anyone asking for it can legally be framed as privacy sensitive information. That would simply be absurd.
The GDPR regulations largely represents common sense and decensy and this über-paranoid consideration about what “may” be covered or not is not really productive use of time.
Example: if you explicitly email someone, according to the GDPR the recipient has been given an implicit right to store your email and email-address. Because there’s no way for them not to. Because that’s just how email and computers works.
I can’t imagine a fucking user-agent string shared by billion of other users enjoys higher protection.
The GDPR is not insane. Chill.