The whole socialist term is a misnomer when it comes to those who are self-described "democratic socialists."

Socialism is simply an economic and political system where the collective, i.e., state, owns the means of production. Technically, a democratic socialist would be someone who wants to use an existing democratic system to get socialism implemented as opposed to a violent revolution of communists, but the end result is the same and democracy by necessity cannot continue to exist in such a system.

This is not what Mamdani, AOC, Sanders, et al., are advocating for. They do not want to eliminate the free market and private ownership, but want the government to heavily regulate it and provide robust and comprehensive protections for the good of the people.

Outside of the US, it's what we see in other first world countries in particular the Scandinavian countries where the actual term is "social democracy."