The ‘classically fascist’ regimes were simply devoted to this doctrine, in the same way classical communist regimes were dedicated to the doctrine of collectivisation.
Their similarities (nationalist, state capitalist, militaristic) perhaps naturally aligned them more with the doctrine of fascism, but this doctrine is by no means unique to any one form of government. You could have liberals, socialists, even anarchists ‘doing fascism’ without being ideologically fascist.