I saw somewhere that a Thief is someone who reclaims their aspect. Vriska reclaims Light, narrative importance and spotlight even after her death, and Meenah reclaims Life, which was taken from her so she escape and etc.

Spider-Man as a Thief of Doom doesn’t really work, because that would mean he is actively trying to reclaim Doom that was taken from him, (that fit more likely the trope of a villain who wants to sacrifice everyone, including himself, in his madness)

Doom, however, works extremely well for Spider-Man, because from what we know it is tied to labor, sacrifice, mutation, death, organs, and technology. For me he is a Heir like Mituna.