I guess I’m mixing up ordinals and cardinals but it seems odd that if you order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2), etc...)you never get to omega and cover all the same points.
EDIT: Just wanted to add that an order isomorphism has two requirements:
(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and
(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).