Okay I know this doesn't really make any sense, but my personal headcannon is that wishes are like a build up of energy the traveler accumulates during their adventure. They use this energy to channel the cosmos and align themselves with the constellations of the friends they meet on their journey. When the traveler tunes into a constellation they can use that characters abilities and weapon arts. As for weapon variety, although the traveler fancies the sword, weapons in Genshin can be easily stored and summoned using magic. Since the traveler has been around for so long, there's no reason they wouldn't have mastered other weapons as well. So they can simply change their weapons and elemental resonance on the fly like it's nobodies business. As for why they can do this? Idk, I'm making shit up.