Cathedral comes from cathedra meaning chair.
Same with the also mentioned Sagrada Familia in Barcelona. I've often seen it referred to as a cathedral but it is just a large church. The actual cathedral in Barcelona is elsewhere. Although now it is deemed a basilica, which I understand is generally just a very special and honored church.
The intention is important, I'm not sure what to call it but "chapel" as distinct from oratory is probably closest. To a Catholic these are the same thing, but: any Christian would recognize this as a chapel, just not a Catholic one, and we have the precise term "oratory" to cover that.
Cathedrals are also usually “alive”, constantly renovated, modified, expanded etc.
If you're desperate to be pedantic about it, you could say this guy tried to build a cathedral. But I think the headline is close enough to being true.
So if I build a city hall in a city, then probably someone might have a problem with that. Because for something to be a city hall it must be built by the municipality or local people or something along those lines. Same with a cathedral - it’s not just a very big church, it actually has a function in the bigger organization etc.
That being said I definitely would not have even clicked the link, read the article, or commented here if it was titled “guy built church” ¯\_(ツ)_/¯
(I do disagree with the GP comment though as maybe this guy was trying to build an actual cathedral. But there's no quote in the article that really indicates either way.)
I just contest that there is only one understanding of what cathedral is.