Thats... hyper anarcho primitivism, taken significantly beyond general anarcho primitivism, not fascism.
Its... accelerationist anti-civilizational hyper-individualism, something like that.
Even anarcho primitivism/ists would say we should return to pre-industrial, tribal societal forms.... this goes even further beyond that and rejects even tribes as social units and units capable.of at least some specialized production... its like a cariacature of anarcho primitivism.
Tyler is the urge to destroy and to feel alive by constantly facing death... given a suave and calculating and attractive embodiment.
You may note how that description you quoted involves... no artificial hierarchy at all, no state, total free for all. It thus definitionally cannot be fascist, fascism requires a state, defined and maligned enemies, jingoism... there are maybe some elements of fascism here, but not many.
Though, his means of achieving this outcome is essentially a semi-fascist cult.
Its also not really about the aesthetic. The leather clothes bit is because leather is durable, civilization has collapsed so you'll have to make your own clothes.
No aesthetic (beyond pure utilitarianism), no hierarchy, no community.