Ultimately, “socialism” comes from Greek philosophy, which has always tried to insert itself into Christianity, so Christianity itself is not the source and the notion actually predates it.
Granted. But nobody today is trying to instantiate the reign of the philosopher kings. Modern socialists are indeed trying to instantiate heaven on Earth though. Their ideologies are always eschatological in nature. They believe in a historical beginning state, then a struggle, and ultimately an ideal end state. There is nothing Greek about that. That's secularized millennialism.