The question is what does one try to achieve in the definition of "cardinal". This is the same as asking whether or not a partial order should be reflexive or not. Both answers depend on what you are trying to achieve.
When the axiom of choice is assumed every set can be well-ordered and therefore all the cardinals are ordinals anyway.
When the axiom of choice is not assumed, or when its negation is assumed [read: can be proved from the assumed axioms] then there are sets which cannot be well-ordered and their cardinal is defined by using Scott's trick.
Cardinal numbers are, in my opinion, numbers which represent the size of a set. We don't always have to have a deep and thorough grasp on how these things represent a size. But we want the notion to be coherent with how we think about addition, multiplication, and even exponentiation. Furthermore equicardinality means that the two sets have the same size.
Now comes the choice of the people working with the definition, which (as I remarked before) is similar to the choice of having zero as a natural number. Do you want cardinals to represent size, or do you want them to represent a well-ordered size? Both options are viable.
If one assumes that only well-ordered sets have cardinals then in some sense non well-ordered sets are like non-measurable sets (and the Hartogs and Lindenbaum numbers are like inner and outer measure). These sets become a pathology, and ignored whenever cardinality is discussed. But there is a big downside, cardinal exponentiation is not defined anymore, and even the real numbers may not have a cardinal. Even more, Cantor's theorem makes no sense because it involves cardinals of arbitrary sets and their power sets.
If one assumes that cardinals can be assigned for every set then one loses the simple and nice definition of addition and multiplication as "maximum". One might argue that infinite sums and products may no longer make sense, but that would be true even in the previous case.
Considering all the above, I prefer to assume all sets have cardinality rather than treating cardinality as a pathology.
To read more:
- Defining cardinality in the absence of choice
- There's non-Aleph transfinite cardinals without the axiom of choice?
- Non-aleph infinite cardinals
- Possible inaccuracy in Wikipedia article about initial ordinals