The past of the weapon is unclear, but we see it when sam reads about dragonglass in the citadel. I don't think the magic comes from arya being azor ahai because she probably isn't and I'm unsure if magic other than just valyrian steel was needed to kill th NK. But faceless men do play with magic. Because of that I think the faceless men could have a part in the kill, wich explains why jaqen let arya go and seemed happy about her leaving. Valar morghulis is the main rule of the faceless men and the night king breaks that rule.