In some set theories such as ZF+GAC, in which GAC is global axiom of choice, the Von Neumann universe $V$ bijects to $Ord$, the class of ordinals. It suggests us that proper classes may also have cardinality，in the example is $|V|=|Ord|$. In addition, if we are in ZF+GAC+ALS, it seems $|V|$ is the only cardinality which is not a cardinal number. Moreover, it seems some properties such as Cantor–Bernstein–Schroeder theorem also holds for cardinality of proper classes, but I'm not sure if it is well-defined and won't cause any paradox...
Answer
There is absolutely no problem with extending the definition of a cardinal to classes, except that we cannot argue within the universe about cardinals of classes as we do for sets. Every argument of the form "All classes such that ..." would be a meta-argument. Of course, one can use a stronger set theory which allows classes, but that's a slightly different story.
Besides the above point, it is not very difficult to prove that Cantor-Bernstein theorem for classes (i.e. the existence of two injections implies the existence of a bijection). And so we can really ask whether or not there is a class function with such and such properties (injective, bijective, etc.)
It is important to note that just as when removing the axiom of choice it is possible that there are surjections which cannot be reversed, without global choice it is possible to have class-surjections which do not have an inverse injection. So it is important to stick to the definition by injections, because that definition works without any use of choice.
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