An important thing to remember about Fascism is that it's... essentially what happens when Nationalism is practically the state religion.

To Fascism, nothing, absolutely nothing can be allowed to supersede the needs of the state or surpass it. Corporations, churches, labor unions, fraternities, and basically anything that you could possibly put before utter devotion to the nation is frowned upon, if not outright destroyed.

It's basically how "National Socialism" could justify cracking down on labor unions. They were not under control of the state, and in spite of being "socialists" (and that's a very, very loose definition) the nation comes before the workers.

I suppose you could think of it like some... pretty disturbing country-wide cult. I don't mean religious theocracies, I mean outright cults, where it seems like everyone is held in the thrall of some "glorious leader", where you're expected to cut out friends and family for "dangerous thoughts" and devote yourself entirely to the cult.

It's also why in spite of being "National Socialists" the Nazis absolutely loathed Communism. The idea of nations and their traditions being torn down, of the lower classes putting aside their differences (especially national differences) to unite together and form one single state, all of that was essentially anathema to the extreme nationalism found in Fascist systems.

EDIT: I'd also like to point out asking questions about Fascism is... pretty difficult.

We only had about two or so "famous" nations self-identifying with this "fascist" ideology. The political ideology is still taboo enough today that you don't really see many self-proclaimed "fascist thinkers" writing large bodies of literature on the subject. Plus there's how it basically became a shorthand for any kind of authoritarianism or "oppression"... aaaaaand lastly a lot of self-proclaimed "fascists" today tend to follow only the National Socialist model of it and even then only seem to be in it for the racist aspects, which wasn't really as heavily emphasized in Mussolini's "fascist" state.

The easiest way to describe it (and thus lacking a lot of depth and complexity when discussing the ideology) is that it combines the hyper-nationalistic, pro-tradition attitude of reactionary conservatives with the "revolutionary" and highly militant rhetoric of radical leftists. The Nazi's economy was structured in a way that can be kind of described as "corporatism" but not in the original definition of that word.

In essence, the most underlying theme of Fascism is utter subservience of all institutions of society to the nation in which they're located. Stubborn, almost zealous use of traditional symbolism while at the same time painting a "revolutionary" view of that history-- if you're wondering what this means... basically Fascists are not Conservatives, they place heavy emphasis on traditional symbolism, but the very history of the "nation" is essentially rewritten to fit a theme park version of it. Both Germany and Italy were relatively "new" nations that were previously clumps of usually warring nation-states and feudal lands that each had their own respective cultures and ideals... however in the Nazis case, they created the myth of some "eternal, Aryan Germany". A mythological nation that "withstood Rome" and "picked up the torch of civilization with the 1st Reich". The "Germany" the Nazis believed in was part myth and part exaggerated historical facts.

So, after the subservience to the nation and zealous use of symbolism, you also have a kind of hyper militant rhetoric. There were "enemies" everywhere, working to sabotage the nation even from within. Good, fascist citizens were to constantly be on the look out for enemies, to fight in any war not because the cause is just, but because every war is a life and death struggle for the continued survival of the nation.

Those are just a few of the most common traits between Fascist nations.

Freemasonry was indeed considered a rival for individual allegiance, along with political parties or trade unions. But the same might also be said of the churches, which lost influence and freedom of action but were otherwise largely able to function. But Masonry was also associated with liberalism, which made it also an ideological bête noire. The liberal or even left-leaning lodges were the first to go, though conservative ones followed.

A few thoughts as a Freemason, myself:

1. Freemasonry arose from the Enlightenment era philosophical tradition. It places a strong emphasis on the value and equality of the individual within the Lodge, and this can be seen as dangerous thinking by those who style themselves above the masses.
2. Democracy is woven into the Masonic system. Though we're not political, we engage in democracy at all levels of our own organization, and just introducing that element in a way that demonstrates its profound functionality can be seen as threatening.
3. Though Freemasonry is not a religion and does not seek to question or modify the religion of its members, the fact that there are trappings of religion present has made some religions very leary of us (Catholicism especially, but not solely) and when religion becomes entwined with the state, this can lead to governmental opposition to the Fraternity as an extension of the religious opposition.
4. The example of the American Revolution is often as source of concern. Freemasonry is often misunderstood to have taken a direct hand in orchestrating that war. In reality, though many individual Masons were involved, many opposed the war. Still, this leads to a sense that Freemasonry involves itself in regime change and can scare any potential or actual tyrant enough to take action.
5. But most simply, the combination of free association and privacy in the Lodge is a source of significant concern if you're worried about your population coming after you for abusing them and their trust.