If you get to know your friend well, you will realize they are unequal in many ways. They're unequal to you and they are unequal to your other friends. Fascism is fanatical nurture fundamentalism, and thus admitting inequalities is heresy. Thus no devout Fascist can know their friends.
Friendship works best if you assume exact social status equality, even though you can explicitly tot up their social merits and demerits and carry out fine rankings. Or rather, you unavoidably do this by reflex. However, luckily, you can waive the obedience rights higher status is supposed to grant, and base the relationship on negotiation and mutual benefit rather than one-sided dominance.
While it's not the core cause of natural narcissism, the core symptom of narcissism is being unable to see others for who they are. Fascists make being blind to others high status, and thus everyone desperately tries to be as narcissistic as possible. Narcissism is a real condition, so the symptoms are related. If you try to mimic one core symptom, you will unavoidably be trying to mimic the whole constellation of symptoms.
We can also get to this through the back door. Fascists believe everyone is identical except for some largely cosmetic accidents of history. Thus, why bother knowing anyone but yourself? If you know yourself, and everyone is identical, you know everyone.
And through a side door. Judgment is discrimination. I hope I don't need to point out how Fascists don't like discrimination. Narcissism is total lack of judgment of character. Gets a Fascist seal of quality.
A true Fascist isn't allowed to know anyone but themselves. A true Fascist approves of narcissism full-throatedly.
P.S. Even the most devout-seeming Fascist often isn't a true true believer. They're aware it's a game. They don't believe the rules, they're just loyal to the rules. However, they are truly loyal. While they approve of performative narcissism, they approve more of natural, genuinely-screwed-in-the-head narcissism. These narcissists will have advantages during promotion, either at work or in advertising campaigns.
P.P.S. For the slower readers: America is a place where you can't really be friends. Americans find genuine affection sincerely off-putting in almost every case. It's a bit weird if it came from nowhere, what with friendship being otherwise a human universal. If it instead it's a necessary consequence of Fascism, it would be weirder if Americans did like friendship. Contrast the behaviour on Friends with the behaviour in any random non-nihilist anime you care to name.
Scene: toked-up sophomores in their dorm room. "Bro...what if...what ifff....like, friends could, uh, like, do nice things for each other... *huffffff* uhhhh... instead of being mean all the time?"
Because Americans can't have friends, they must have a nanny State or they would be entirely without support. The State, of course, is only too happy with this state of affairs. P.P.P.S. Americans are anglicized, so they can't have families either.