Question: There are definitely at least two ways in which the word "are" is used. Consider the following:

> 
> Book A and Book B are red.
> 
> 
> 

vs 

> 
> Book A and Book B are similar. 
> 
> 
> 

In the first use, it is applying an adjective independently to multiple objects. You can view it as syntactic sugar for 

> 
> Book A is red and Book B is red. 
> 
> 
> 

However, the second sentence doesn't allow such an expansion. 

Now, I come from a CS and math background, so my view of this is a bit theoretical (to me, it seems that there is a difference in composition between the two example sentences). Grammatically or linguistically, is there such a concept as the difference I have described?

Answer: 
> 
> [1] *Book A and Book B are red*.
> 
> 
> [2] *Book A and Book B are similar*.
> 
> 
> 

Yes, there is a difference. 

Example [1] is called **distributive** coordination, while [2] is called **joint** coordination. 

In [1] the property of being red applies to book A and book B separately -- it is distributed between them; whereas in [2] the property of being similar applies to the two books jointly, as a group.
The flaw in your question is assuming the answer has anything to do with the properties of the verb *to be*. 

Rather, it's your choice of adjective that is not allowing you to use the transitive property on these sentences. 

Your first example uses a simple descriptive adjective. *Red* It requires no comparison to work. Hence, the logic:

> 
> A *is* C. B *is* C. A & B *are* C.
> 
> 
> 

Your second example is a different type of adjective. It's a comparative adjective. The logic still applies.

> 
> A is similar. B is similar. A & B are similar.
> 
> 
> 

But, because you used a comparison you're left asking *similar to what?* if you view it as two singles.

You'd have similar issues with any comparison. Taller, greasier, stinkier, uglier, etc.