Political Memes

As I understand the terms (which may well be wrong), you're correct and Denmark would be socialist. There might be some distance between "full" socialism wherein all public services are owned and controlled by the public (= the state), but the direction of social policies seems about right.

However, particularly in the US, that distinction has been smudged into one collective "evil leftist" scarecrow. I'd assume the person in the OP either doesn't know or consciously accepts that fudging to make a case for why their system is beneficial rather than the "scary socialism". Deflecting from the label to a descriptor (particularly in the context of profit-oriented mentalities, where "good accounting" would be desirable) dodges the preconceptions and propaganda attached to the former.

And ultimately, I think that's more fruitful for explaining an ideology. Starting with a label that carries connotations primes expectations accordingly. Starting with and focusing on the benefits invites engaging with the topic and creates a different foundation for productive conversation.

What's in a name? That which we call a rose
By any other name would smell as sweet. So ~~Romeo~~ would, were ~~he~~ it not ~~Romeo~~ Socialism called,
Retain that dear perfection which ~~he~~ it owes
Without that title.

I'm not gonna suggest we stop calling socialism by that title, but I do see a case that aome discussions should be led with descriptions and arguments instead of terms that have been vilified to the point of stifling any reasonable discourse.