Does the usage of mathematical symbols work differently in books than in theses?

So why not just define a tau with a subscript v and a tau with a subscript r - make it clear in your nomenclature and you're sorted.

See:

From what I know, one cannot use a mathematical symbol for two different notations in a thesis. Is this right?

That's certainly good advice, since it avoids unnecessary confusion.

The thing is that I am trying to cite a book in my thesis, but this book uses (tau) for both viscous stresses and reynolds stresses. Can I do the same in my thesis or do I have to choose another notation?

Presumably you don't intended to use τ to mean viscous stresses and reynolds stresses in the same context, because usage of τ would then be ambigious. So, I presume you'll be using τ to mean viscous stress in one context, perhaps an entire chapter, and to mean reynolds stress in another context, perhaps a different chapter. This seems reasonable. However, you needn't use the book's notation, especially if there is a risk of ambiguity.

From what I know, one cannot use a mathematical symbol for two different notations in a thesis. Is this right?

Such things are not governed by hard-and-fast rules. Your institution might have specific style guidelines for a thesis, but notational choices are usually at the discretion of the author. Of course, it's ultimately your advisor and/or committee who decides whether your thesis is acceptable, though it's unlikely that notation alone would be the cause for rejection.

The thing is that I am trying to cite a book in my thesis, but this book uses (tau) for both viscous stresses and reynolds stresses. Can I do the same in my thesis or do I have to choose another notation?

It's up to you. Make your own decision as to which way will be the most clear. You can ask your advisor if you are unsure.