“Socialist” is an arbitrary word and can mean just about anything. There have been democrats in the past (like President Truman) who would have never used the word socialism but who supported the same kinds of policies as Sanders.
There is no fundamental difference between America and Europe when it comes to the welfare state, labor market regulations and health system — it’s just a matter of degree, of where we fit on the spectrum or continuum. America has a welfare state, it’s just small and neglects the poor (though things like the EITC and food stamps really do help to decrease poverty by quite a bit).
It’s just a matter of confused terminology. People on the center-left in America are called liberals or progressives, in UK and Ireland are called Labor, in Germanic countries are called Social Democrats, and in Latin countries are called Socialists. Each country is a bit different and going in a slightly different direction….Scandinavian countries have the biggest governments but are shrinking them a bit, and we and the UK and Ireland have the smallest but are growing them a bit. But we’re all on the same continuum. It’s not as if there’s some magical line so that if you go from having 45% of the economy be the government to 46% you’re all of the sudden socialist rather than capitalist.
Bernie calling himself a socialist really just means he wants to go somewhat farther on the spectrum toward a welfare state and more economic equality. So all Sanders would do differently is things like progressive taxation (like under Eisenhower!), Canada-style health care (which Truman advocated and which old people already have in America, that is Medicare!), making it easier to join a union, creating government jobs for to ease poverty and create full employment, and perhaps getting rid of trade deals that bleed American jobs to China and Mexico.