MRS variables are logical variables. This means that they specifically stand for individuals in a model structure. As you say, handles are mathematical variables, in the sense that each handle must be equated with a label in a fully scoped MRS. However, a label/handle only ever refers to a linguistic object and never an individual in a model structure. This is really important if we want a well-defined logic, where we can have a model structure to represent a situation, and an MRS to represent a sentence, and then evaluate the MRS on the model structure.
However, for many applications, we don’t need a well-defined logic and so the logical variables can seem cumbersome. Hence the motivation for a simpler variable-free representation.
As for the specific quote, I think it refers to a situation where there are two analyses for a sentence but they share the same analysis for a particular NP. In assigning a unique identifier to each variable, the identifiers might end up being different because the analysis of the rest of the sentence is different, even though the analysis of the NP is the same. If the identifier instead comes from the token, it will be the same identifier across all analyses.