First and foremost the "fascism is socialism" crap is so widespread because it is convenient to the ruling class. It is one way in which you can justify having a safe centrist ideology that will be socially acceptable and not get you in trouble with powerful people.

It is no coincidence that there are these superficial similarities, fascism was developed as an antidote to socialism, a form of inoculation that would make the same kinda of appeals while being completely harmless to the bourgeoisie.

Like all other good pieces of propaganda it has a kernel of truth, while being fundamentally different and opposite ideologies, fascism and socialism shares some superficial differences that are easier to spot for the historically and politically illiterate than the deeper structural changes that actually matter.

Fascism and socialism are both revolutionary in rhetoric, saying that the old system must go. This puts both in opposition to the liberal-conservative mainstream. But while socialism is revolutionary in praxis, fascism is revolutionary only in aesthetics while being materially conservative.

Another superficial similarity is the appeal to the downtrodden masses, both makes promises of bread and dignity. Again, this puts both socialism and fascism at odds with the mainstream who openly detest the people. While socialism is actually about delivering on those promises, fascism is all about entrenching the class rule that leeches off the people while only delivering on those grand promises to a small in-group.