So Oath is not a legacy game, because there's no permanent changes to the game (no destruction, no stickers, no writing, nothing). It's not a campaign game either since there's no overarching narrative covering multiple games (well, not one provided by the game, at least). So it's kind of its own thing.
I really like the idea of what Oath is going for: a living game that changes and evolves with each play, but not in a permanent fashion, and not with an end. In that sense it's markedly different from a legacy game (which has both permanent changes and also a set end to those changes). But when trying to find other games like it I find that I don't have a word to describe it. It seems like right after Risk: Legacy came out everybody agreed on the legacy tag for that kind of game, and then when Pandemic Legacy came out it was irreversible. Now everybody knows what a legacy game is. Oath seems to be doing something just as new as the legacy thing was back then, but no term seems to have come out. Like, there's no category of "chronicle games".
I'd still call it a legacy game. While legacy games usually involve physically, permanently, altering the pieces, it's not required; just that previous sessions carry some impact over to the next. At least that's what I've seen.
I would say simply adding and removing cards from the game counts as a legacy mechanic, since at any time my copy of Oath is going to play different than your copy based on our play history.
Maybe "legacy-lite"?
Edit: seems the term might be "Fable"? https://boardgamegeek.com/boardgamefamily/44081/series-fable-game-system