It doesn't really matter for people who don't use magecraft. In the case of Arturia, I would say that Avalon, and probably also her swords, are what bestowed her with magical powers. Being fused with Avalon is also what allows Shirou to develop his powers, and is the origin of his 'sword' affinity. Arturia originally was just a powerless girl who wanted to help her country. After picking up the swords and Avalon, she became their vessel. If I had to guess she has the 'sword' affinity like Shirou.