It's legally structured as a private company, but it's really not. It's 60% owned by the Swedish state and 40% by the Danish state.
Ownership by states doesn't make it a public service. It makes it a company whose shareholders happen to be governments. It still operates under corporate law, not administrative law, and it was explicitly removed from being a state service.
Calling that "not really private" is just rhetorical framing, not a legal or operational distinction.
I'm using it in the legal/operational sense relevant to whether a government service was withdrawn (agency vs company, public law vs company law). By that definition, it's a private company.
You're using it in a shareholder-ownership sense. That's a valid perspective, but it's a different question than the one being discussed here.