Arendt is misidentifying what is going on, broadly.
There is no similarity, even in the most extreme functionalism, for anything like her shoe analogy. Her shoe fulfills other functions as well, and fulfills them better, so it is not functionally equivalent.
So are you.
No one paying attention to function, after identifying both red and green as colors, would declare them the same. One signals ripeness of apples, the other, their unripeness. They serve different functions.
Beyond that, there remains a difference between using definitions in terms of intention and roles, and demanding that this is the only proper way to define things.
We can posit deeper aspects of the definition of a chair, but refusing to use the definition in terms of being intended to be sat upon primarily by an individual in the meantime is pretentious silliness.
And one need not get anywhere close to even that standard, to consider Communism religious. One just needs to look at how philosophy deals traditionally with the belief systems of other cultures. Many versions of Buddhism and virtually all forms of Taoism and Confucianism are as atheist as Communism. But we in the West have decided to look upon them as religions.
To exempt Communism from the same label, it would need to have a distinction from them, with a difference. Simply declaring itself atheist is not enough. These other religions have variants that explicitly declare themselves atheist, and a range of popular interpretations that disdain the question of theism altogether, as pointless.
And any form of prophesy that is based upon a specific belief that does not bow to material contradiction, and compels some degree of action, seems only to fit into the realm of religion.
This observation is not a "functionalist" one. It does not lay down a theory of the intentions and roles of religion. It only links one similarity to others inductively in a relatively straightforward way that is not tied to any specific rigid formula for definitions to take.
It seems to me that there is a whole range of options short of rigid functionalism on which one should reasonably identify many strong political philosophies with 'magical' components as religious.
To my mind, these include certain forms of racism and nationalism, which involve origin or foundation stories that are clearly mythologized, and such things as the belief in trans-gendered people, which involves a definition of gender that is completely immaterial. These notions tend not to aggregate up into a whole religion that can stand as a basis for a style of life. But they are religious in form, and when they are assembled into a system that assigns life a purpose, at some critical level, that clearly becomes a religion.
This whole middle ground could also take or leave essentialisms of different varieties. So these two cannnot represent a clean division, where the absence of one necessarily presumes the other.