Given a model of ZFC, is it correct to talk indistinctly about cardinals and initial ordinals, namely, ordinals $\alpha$ such that for every $\beta < \alpha$, there is no bijection between $\alpha$ and $\beta$?
In particular, can we write $\omega_n = \aleph_n$? This I think is correct but the literature often seems reticent to do it, and would rather address $\aleph_n$ as the cardinality of $\omega_n$.
If you think about ordinals, you'll remember that those are just canonical representatives from the equivalence classes of well-orders under the isomorphism relation.
The finite ordinals and $\aleph$ numbers can be seen as canonical representatives from the equivalence classes of well-orderable sets under the equipotence relation.
Both these approaches are useful since they allow us to talk about the class of all ordinals, or all cardinals. If each cardinal, or ordinal, is just an equivalence class as above, then those will be proper classes (except for $\varnothing$, anyway) and then there is no collection of all cardinals or the collection of all ordinals.
Of course that it is good to separate the two notions, since $\aleph_1+\aleph_0=\aleph_1$ as cardinals, but $\omega_1+\omega\neq\omega_1$ as ordinals.