That's the part that's "weird" in the sense of being very different from the similarly-named concept in classical mechanics of velocity vector magnitude. But m

That's the part that's "weird" in the sense of being very different from the similarly-named concept in classical mechanics of velocity vector magnitude. But moreover, the rest mass isn't the issue here. Momentum in classical mechanics also takes mass into account but doesn't have the "weird" property that two objects of the same rest mass always have the same momentum magnitude.