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.
Both "are" and "is" are tenses of "be", and "be" can mean several things. The definitions of "be" (from [Merriam-Webster](https://www.merriam-webster.com/dictionary/be)) we are concerned with is:

> 
> D: to have a specified qualification or characterization
> 
> 
> 

Your first and third examples are:

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

This is the English language equivalent of the distributive property. If Book A and book B are red, then I can distribute the verb and adjective to both while conveying the same meaning. Instead of a compound subject, I now create a compound sentence. This is exactly what you describe in your question.

Your second example is:

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

"Similar" describes a relational characteristic. You cannot distribute "similar" to apply to each noun individually (without additional context) because it requires two objects to form a comparison.

To explain in CS terms, think of it as "are" being a token that relies on the following token to define its own behavior.

Certain following words may define characteristics of each of the preceding nouns:

> 
> These books are expensive.
> 
> 
> 

Others may define characteristics between each of the preceding nouns (

> 
> These two are lovers.
> 
> 
> 

Lack of a following word may describe its inclusion in a previous statement

> 
> Which books are on sale?
> 
> 
> These books are.
> 
> 
>