Mussolini hated Hitler, but over time, had no choice but to support him as his armies faced troubles in the Balkans. But true, his rule was much more benign that Hitler.
And it’s worth nothing he started out as a Socialist.
Fascism is a very odd ideological construct.
While it has much in common with Marxism and other forms of socialism, it differs wildly in other ways.
Whether Fascism and Nazism are “really” socialism depends on your definition of socialism. Settle that and then we can talk about whether they fit it.
To my understanding, socialism as a broad category is all forms of ideology with human equality as their goal, taking the legal and social equality of the American Declaration of Independence and adding to it the goal of enforced economic equality. A step too far, IMO. Most forms of socialism have universal human equality as their (claimed) goal, cutting across national borders.
Obviously no socialist system to date has come anywhere close to achieving that goal. Many have increased human inequality, in both social and economic senses, and some of these states have committed the greatest crimes in history in (supposed) pursuit of this goal.
Fascism (the capitalized Italian version) OTOH, was an attempt to mobilize one nation, the Italian, vertically, rather than all workers in the world horizontally. A very different kettle. They mixed it up with various weird ideas of the Nation and State as an organism, which implies different parts of the body having different functions and therefore not being really equal.
But the emotional pull of Fascism was very largely based on its being opposed to the (internationalist) socialists. Fascism was for most of its career national and cultural, as such, not racist.
Nazism, OTOH, was also internationalist, but with Germans and eventually all Aryans of the world uniting against the lower races. It was specifically racialist and racist, not just nationalist or culturalist. It thus rejected the idea of human equality, much more directly than Fascism, and so was utterly anti-socialist, at least if you accept my definition of socialism as reasonable.
All this is the theory behind these various totalitarian movements. Their actual function and oppressive nature varied a lot less.