Because people think they are rivals or something like that? Maybe also because of story reason.

But how are they rivals? One is truth seeker and other is criminal mastermind, how do they even intersect?

In the original stories about Sherlock Moriaty is the one who stood in his way and 'killed' him, or at least Dr. Watson pretty much assumes Sherlock and Moriaty both died after their final encounter when they fell down the Reichenbach falls without their corpses being found afterwards. Moriaty as a character had only minor appearances in the books and short stories. However, according to Arthur Conan Doyle himself he only wrote the apparent end of Sherlock like this because he grew tired of writing more stories about him despite Sherlock's popularity back in time already and needed to get rid of his own creation to go on writing other stories. At a later point Doyle claimed Sherlock survived while only Moriaty died and Sherlock only faked his own death to conceal his existence from other dangerous rivals. Moriaty isn't Sherlock's only one or greatest rival but he is the one who drove Sherlock closest to his near-death.

You kinda answer the question yourself.

Its because Type-Moon perceived them as rival that people will not questioned similar to Naruto and Sasuke; doesn't matter if it doesn't make sense, no explanation, just like how BBC Sherlock made them maniacs disregarding their OG background for modernization because surely people knew they're famous as archenemy to each other.

In original Doyle works, Moriarty is just the 'big boss' intended to wrap up Sherlock stories as Doyle felt little to no interest working on Sherlock--until he got financial issues that forced him write Sherlock again.

Personally, I think the best iteration of Sherlock and Moriarty relationship is the one in Robert Downey Jr. Sherlock movies. Moriarty is the smartest person among nobles of England, for him to be able to establish perfect crime that shaken social structure is testament of his intellect. Sherlock OTOH is the opposite, he's got pride as someone helping others with his brain--contrast to his brother, Mycroft.

For Moriarty, Sherlock is perfect because thanks to him he can constantly push the limit of perfect crime under his job as college professor. For Sherlock, Moriarty is perfect because the world is boring without something to challenge his brain.

Because of the story of Sherlock Holmes. I assume you have never read the Sherlock Holmes books.