Socialism is a fraught term that doesn't actually refer to anything at all. It's like the label "Fascism", which also doesn't refer to anything at all. Not even the so-called "Fascists" could figure out what "Fascism" is. Some of you may recall that Benito Mussolini convened an international conference with the intent of forming a "Backintern" (Black International), as a "Fascist" counterpart to the Comintern (Communist International, also known as the Third International). The conference completely fell apart because the various Fascist parties couldn't agree on much of anything and couldn't figure out what "Fascism" is. A notably absent party to these talks was the NSDAP (also known as the Nazi Party), which had developed the intensely Darwinistic/Dawkinistic, race-based ideology known as National Socialism.
People who have read about Mussolini or read Mussolini (like his diaries) know that he had no coherent ideology. He didn't have anything even approaching a coherent ideology. He was not an ideologue, but rather an opportunist who was totally fucked by the dual superstructures of the RCC (Roman Catholic Church) and the Italian Monarchy. The only way Mussolini would have succeeded was to destroy the RCC, behead the monarch and abolish the monarchy, and importantly, refusing to ally with Germany (beginning with the Spanish Civil War, in which they joined forces to support the psychopathic Franco).
As for Communism and Anarchism, the end goal or end-state (pun intended) is exactly the same - the abolition of the state, to have no state at all. The end of the nation-state (or state-nation), the end of nationalism, the end of borders, the end of markets and exchanges, etc.
The only difference between Communist/Marxists and Anarchists is that they disagree on how to achieve this objective. Do they use fire to fight fire (in other words, use violence to terminate the State's monopoly on the use of violence and ultimately the State itself) or not?