Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets (amorphous sets), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers here much more succinctly.

Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses?

Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of transfinite function composition which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.