Why Do Structured Sets Often Get Referred to Only by the Set? Why do structured sets, like (N, +) often get referred to just by their set?  Under this way of speaking, where N denotes the natural numbers, + addition, and * multiplication, (N, +, *) and (N, +) both can get referred to as N.  But, due to our ancestors we can readily talk about (N, +) via Presburger Arithmetic, and (N, +, *) via Peano Arithmetic, which readily makes these structures different, since equalities in  (N, +) can get decided algorithmically, but they can't for (N, +, *).  But, the way of referring to these structures by the set N masks all of this.  So, why even bother referring to a structured set by its set in the first place?
 A: Because people are lazy there is value in lossy compression for the sake of communication. I agree that this can be a bad convention in the sense that it can create confusion, but 1) usually by context you can tell what structure is assumed, and 2) sometimes people want to consider multiple compatible structures without explicitly listing them, which are again usually clear from context. 
A: It's a question of brevity, for the most part.  Brevity is different from laziness, because brevity has the goal of clarity.  In theory, we could require that all our proofs and writing in mathematics be so rigorous that a computer can read it, but then it would be unreadable by humans.
So, a news article will refer to "Secretary Clinton," or even "Clinton," perhaps only once referring to "Secretary of State Hillary Clinton." The reason is that humans are very good at determining context and meaning, and they find redundancy leads to confusion in communication.  (This is why we use the word "it" in place of nouns, too, and that can cause confusion when misused, as can referring to "Clinton" if the article contains information about both Bill and Hillary.)
So, if the context isn't clear, then a person should definitely write $(\mathbb{N},+)$, but it's not always obvious when the context is clear or not.
A: It is a general convention in mathematics that almost any structured object is primarily a set, with the structure as a sort of decoration.  For example, the sphere $S^3$ is primarily the set of points in the sphere, and the various other metric, topolgoical, and algebraic structures on the sphere are considered secondary.
In the specific cases you have mentioned, the phrases "Presburger arithmetic" and "Peano arithmetic" refer primarily to specific first-order theories within the context of logic.  I'm not sure why referring to a structured set via its first-order theory would be any more or less natural than referring to a structured set via its underlying set.
