In his book Basic Set Theory, Levy wrote:
The full associative law and associative-commutative law can be proved only by means of axiom of choice.
(page 104 in the Dover edition)
Here Levy is referring to infinite sum of well-ordered cardinals (a.k.a. alephs), which is defined as follows:
$$\sum_{i\in I} \kappa_i:=\left|\bigcup_{i\in I}\{i\}\times\kappa_i\right|$$
I don't know what version of the associative-commutative law Levy has in mind, but one is as follows:
$$\sum_{i\in I} \kappa_i=\sum_{j\in J}\sum_{i\in X_j}\kappa_i$$
where $\{X_j :j\in J\}$ is a partition of $I$.
I can't see how this uses the axiom of choice. One could say $\sum_{i\in X_j} \kappa_i$ might not be a well-ordered cardinal and thus the sum on the right hand side is not defined, but I could view this as a statement purely about existence of bijections between two sets, and the proof is choice-free. Am I missing something obvious or Levy has other versions of the general associative law whose requires the axiom of choice?
The problem is not that the sum is by definition the cardinality of the union. It is whether or not this definition really holds up to cardinality.
The cardinality of $\sum_{n<\omega}2$ should be the cardinality of any countable union of pairwise disjoint of size $2$. Not just the cardinality of $\omega\times2$. But without choosing bijections, we cannot guarantee that this statement holds. For example, there could be a proper class of pairwise incomparable cardinals, all of which are "countable union of pairs".
And even if you insist on staying within the limited range of ordinals. Remember that a countable union of countable sets could have size $\aleph_1$.
You might also be interested in this MathOverflow question and the answer I wrote to it.