There's obviously different theories of what fascism is and how to identify it. I am most sympathetic to the one outlined their, which is essentially it developed an independent pole of attraction tht had a paradoxical position of doing aspects of what the elite wanted, but also existing as a threat in other areas. The social bae of the Nazi's (and indeed Mussolini's Fasci) were lumpen thugs and aspects of the middle class, at the lower end who felt "squeezed" above and below.
LFC have a very similar dynamic in their support base. Lots of home counties types with range rover and a subscription to the Telegraph and the lumpen fools who try to torch iconic symbolsof a working class city, or urinate on memorials/graffiti them. Go and watch Tommy Robinsons visits to Liverpool, the crowds that go to watch him are full of LFC shirts. They're a racist club, with racist support (he himself went to Anfield, and no harm came to him, I suspect he was welcomed).
As for the 2nd part, them and United are the establishment clubs. They're as establishment as Saville and Thatcher. Loads of tories support them, including now Rees Mogg. They are as English/British as the Union Jack. They are trying to colonise a working class city with their nonsense as well. They'll lap it up being on the Proms as well. There will be absolutely no calls for the club to distance itself from the establishment once YNWA becomes the new song of british imperialism at the Proms. Most of them will be dead chuffed they can asscoiate themselves to the mass murder of the British empire, as they hate Black people and Irish people.