I agree with Jenipar. This is not a commonly used expression - it is language from a piece of creative writing or fiction, I am guessing. Think of a paper napkin all wet - it becomes like pulp, or "soft, wet, and shapeless material." If you cried very much and used the napkin to wipe your tears, you would basically moisten the paper napkin so much it becomes like wet pulp. So the expression is just a very creative way to say that person "cried very much."

This is a very descriptive and poetic phrase. People would usually not say this, and you'd get strange looks if you said someone did this. I'm not sure what specifically a Japanese napkin is, but the idea is that she cried so hard, dabbing her eyes and maybe wiping her nose on the napkin, that the napkin became destroyed, an unrecognizable mess.

Yes it's just the author's amusing 'turn of phrase'