Bernie does call himself a "socialist." But I don't think it's truly accurate for him to do so. Although colloquially, in the American context, and certainly in right-wing usage of that term, he is. But he doesn't want government to own the means of production, and he's a pretty average left-person by European standards.
As for health: I didn't touch it this time. Complicated subject; it's not fully centralized in Denmark, but Denmark certainly has more government involvement in the health sector than we do. https://en.wikipedia.org/wiki/Healthcare_in_Denmark
But what really sells me on single-payer or other significant healthcare reform? Denmark and other countries spend around 10% of GDP on healthcare. We spend about 20%. And in most cases they actually have better health outcomes than we do.