Is is possible for a Heart player's Splinters to have a class/aspect/full classpect different from their original counterpart, should they even be able to possess one? For example, if one were able to separate parts of their soul or something–or some other form of Splintering.

Splinters are a Splinter of the same Soul, unless something extremely dramatic happened they’re still the same Soul, and as such, the same Classpect. Not to say- The sense of Self of a Heart Player doesn’t always have to be Splintered, Splinters are kind of specific.