[cokernel]

Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.

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).