Uh, right. Fascism recognized the failings of Communism in destroying old orders and institutions, and sought instead to co-opt them under the State. And justified its totalitarian collectivism via the Nation rather than by the Class.
Thus the bundle of sticks.
As such, Mussolini and Hitler (and FDR) didn't do away with or nationalize corporations, unions, the church, etc. but instead decreased the number (in Hitler's case this was often down to "one" because of his paranoia) and increased the size of the institutions so that all the "important powers" of society could meet and enact the "best path" for the State.
Make no mistake. A corporation wasn't free to produce whatever it wanted on terms it wanted, it had to follow State diktat, but unlike in the Communist countries, the corporation was still free to profit and obstinately existed in "private" hands. The fascists recognized that the individual power of institutions including the corporation but also the union and church and social groups were strong unifying forces that maintained efficiency much better by "operating under their own ways" to enact State policy rather than also being managed by the State. This is why they were much more successful than the Soviets in not killing their citizens trying to improve the economy. (Hitler was then free to kill them for other more rational reasons.)
The institutions went along because something is better than nothing, and eliminating all your competitors is a small price to pay for being even further entrenched as an elite class or group. The alternative was the State crushing them and replacing them.
In fact, one of the very institutions so important to the Fascist regimes was "the law" which is why Hitler spent so much time getting t's crossed and i's dotted when he likely could have seized power much earlier by toppling the Weimar regime in a coup. (And why "just following orders" wound up being the defense used...it was the law.)
Citizens United, campaign donations, lobbying, and other aspects of the freedom of the press are in opposition to fascism because they provide avenues for inter-corporate competition. The fascist state controlled the press and prevented inter-corporate competition because inter-corporate competition allows corporations to pursue their own ends instead of the ends of the Nation which was the single priority and duty of citizens in a fascist state.
Alter is claiming that corporations and citizens owe a duty to the State, namely America, and should be signing "loyalty oaths" to it where they pledge to act only in the interests of the State with an explicit threat that anyone who doesn't Pledge Allegiance will be considered an enemy of America. This, on the other hand, is pretty much fascism.
EDIT: For some reference, from your own Wiki quote:
Fascism's theory of economic corporatism involved management of sectors of the economy by government or privately controlled organizations (corporations). Each trade union or employer corporation would, theoretically, represent its professional concerns, especially by negotiation of labour contracts and the like. This method, it was theorized, could result in harmony amongst social classes.[31] Authors have noted, however, that de facto economic corporatism was also used to reduce opposition and reward political loyalty.[32]
In Italy from 1922 until 1943, corporatism became influential amongst Italian nationalists led by Benito Mussolini. The Charter of Carnaro gained much popularity as the prototype of a 'corporative state', having displayed much within its tenets as a guild system combining the concepts of autonomy and authority in a special synthesis.[33] Alfredo Rocco spoke of a corporative state and declared corporatist ideology in detail. Rocco would later become a member of the Italian Fascist regime Fascismo.[34]
Italian Fascism involved a corporatist political system in which economy was collectively managed by employers, workers and state officials by formal mechanisms at the national level.[35] This non-elected form of state officializing of every interest into the state was professed to reduce the marginalization of singular interests (as would allegedly happen by the unilateral end condition inherent in the democratic voting process)
EDIT2: Fascism lost The War but basically won the war.