This sounds as if we are talking about sharing one object with
I disagree. I think the meaning is really clear. But if you really want to avoid using "share" (which I wouldn't), you can say simply:
B doesn't have the same fascination for reading and writing as (does)
(does) can be elided completely or placed before or after A.
or using "share":
B doesn't share A's fascination with reading and writing.
If you really want to use the structure "neither" "nor", then:
Unlike A, B has a fascination for neither reading nor writing.
Unlike A, B doesn't have any fascination with reading or writing.
or more loosely (allowing more leeway from your original sentence):
Unlike A, B is uninterested in reading and writing.
Unlike A, B is not an avid/enthusiastic reader and writer.
or quite longer and unnecessarily wordier:
B doesn't have the same interests as A insofar as B doesn't enjoy
reading and writing.
or separate completely:
A has a fascination with reading and writing; B however doesn't.