Communism is a totalitarian ideology - advocating for suppression of individual freedoms in the name of common good (aka in the name of "people"). One may agree or disagree whether it is a good idea, but it is not consistent with liberal democracy practiced in most western countries.
To add material from comments:
what is singled out is not how party is perceived (as the OP suggests), but the ideology it represents. A party that has "communism" in its name cannot pretend that it has nothing to do with the works of Marx, just like a party that calls itself fascist or national-socialiste cannot claim that it is unrelated to its infamous historical predecessors.
Why is Communism singled out?
Communism is not singled out - rather it is cited as an obvious and notorious example of totalitarian ideology, the primary target of the law. The clause or any other totalitarian party is added for inclusivity, to account for more obscure or yet unknown ideologies (fascist and national-socialism had been dealt separately a decade or two
Are all Communist parties seen as being the same (there is a huge difference between, say, the French Communist party and the North Korean one)
As I pointed out above all the communist parties follow the same ideology. Contrary to laymen opinion, communist parties in 50s were not just political parties like all others - these were paramilitary organizations, aiming at clandestine preparation of a revolution (i.e., overthrow of the existing political order), subject to the strict party discipline, and to orders from Moscow (nor simply financial support). Thus, La Partie Communist Française, mentioned as an example, refused to condemn the German invasion of Poland in 1939, continued to blame the war in the "British and French imperialism" even in 1940, when the Paris was occupied by Germans, and praised Marechal Pétain for concluding the armistice - for during this period Nazi Germany was tied to the USSR by the non-aggression Molotov-Ribbentrop agreement, and the party was directed to support the Soviet line. Only after the German attack on the Soviet Union did the party join La Résistance, becoming a leading force in it - partly because it had already developed paramilitary and command structure and partly, because de Gaulle allowed it to take the leading place in exchange for the Stalin's support for his leadership (unlike various governments in exile residing at the time in London, de Gaulle did not officially represent any government, and there were alternative candidates for leading French, notably General Giraud.)