Obviously the Empire thought so, at least at the beginning and middle of the journey. The character hit all of the macro-level characteristics of the hero, but it was wholly an Empire plot, no? Azura says the character has been chosen, yes, but she also stood a lot to gain by pushing this fabrication along. Ultimately, I suppose it really does not matter if the Nerevarine was real or fake given that Dagoth Ur was ultimately defeated. The complexity of the plot, though, is one of the reasons MW is easily my favorite TES game and one of my favorite games of all time.

Not Indoril Nerevar himself.

However the existence of Azura's champion. It's because the Nerevarine didn't serve the Empire nor did he/she stay around to cement the foundation of their existence or their achievements. When you go missing. People eventually forget you exist and it's not like the Nerevarine told anyone where they were going. Some say he went to Atmora. But he could have left that region within the time of Alduin and the LDB. There's really no telling.

I'd say that the Nerevarine became a legend. Much like some religious figureheads that people will argue the existence of.

Neither did the Hero of Kvatch who is rumored to be New Sheogorath...possibly erasing the Hero of Kvatch's existence. The Dragonborn met a similar fate, but didn't mount a god like the HOK, The LDB just became an immortal champion taking Miraak's place.

Nah, the Nerevarine was real, it's the player who was the fake. Contrary to popular belief, that "Reality" thing doesn't really exist.