Alex's "AMP Factor" is actually an early generation Quirk.
The first Quirk, according to My Hero Academia, awakened in China, so it also stands that a Quirk could awaken elsewhere outside of there at some point. Alex Mann's Quirk wasn't created by the government, but in reality was awakened by accident by the project. However, because it wasn't entirely intentional, Dr. X and Simon understandably only understand how it works to a certain extent, hence Dr. X's fixation on trying to fully understand and obtain the power for himself. Coincidentally, this is also part of the reason why no one else is trying to destroy Alex for having this Quirk; not only is the AMP Factor Quirk's existence basically top secret, but because of the ties to the government, it was mistakenly seen as the result of the project rather than the random manifestation of superpowers it really was. Additionally, AMP Factor's powerset tends to be very subtle, especially during the first season, which would make it very hard to tell just from looking. Even the differences in Season 2's Amp Factor could be attributed in-universe to Alex Mann's ridiculous athleticism and skill as an extreme sports star, thus providing him with the perfect cover.
