How to say one person does not share another person's fascination with either of two topics (using "neither...nor")?

This sounds as if we are talking about sharing one object with another.

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) A (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.

or simpler

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.

or

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.