The "engineer" part usually starts a discussion because in some countries it's a protected term so software engineers can't call themselves that. Or some people think that because the word engineer is in the title, people with the title should be held to some higher standard or somehow regulated - this isn't a thing anywhere for software engineers as far as I know.

> Is being a software engineer a state of mind or an actual profession that comes with certain prerequisites?

It's a job title. If your job has the title of software engineer, then you are one. The rest is either stories that you're telling yourself ("I do not consider myself a software engineer") or bullshit gatekeeping ("you're not a real $x until you do $y"). It doesn't make any difference whether your title is software engineer or software developer.

Titles aside, if you have a problem with confidence, spend more time learning and doing difficult things.