I don't understand. Speaking of a "vector" presupposes that you have a vector space in mind, and the definition is subordinate to that. Of course, that vector space is also an abelian group, and it is also a pointed set (with the origin as the "point") and it is also a set so your vector is also a group element and an element of a set (or pointed set). Where is the problem?

A person may be a "registered student" in one class, and an "auditor" in another class. The person is the same person. So why is it that the same person sometimes is a "registered student" and sometimes they are not a "registered student"? Aren't they the same person? The answer is "context." "Is a vector" is shorthand. The vector space is generally understood from context.

It is informal notational abuse, just like saying $\,ax^2+bx+c\,$ is a polynomial. Usually there is abundant context that quickly helps (humans) infer the precise denotation.

I find your suggestion unnecessarily cumbersome and pedantic.

There are some contexts where such precision is needed, e.g. model theory and universal algebra, e.g. see Wilfred Hodges on Naming of parts, e.g. see the structure superscripts on the distinguished constants of the structure.

Instead of choosing $(0,0)$ as the origin instead choose some other origin $(a,b)$ and then define your vectors as $v=(x,y)-(a,b)$. So there is nothing special about $(0,0)$ as an element of $\mathbb{R}^2$ and we can pick any element we like to be the origin. So to create a vector space we have to add some additional information, which is a choice of origin.

@CyclotomicField It seems you've missed the point of the question.

Things are defined by what they do, not by what they are.

@Gerry In this context (elements of an algebraic structure), it is not what they do but rather what is doable to them, i.e, how the elements are related to each other under the operations of the structure, i.e. structures with the same operation tables (up to order) are isomorphic. See here for more.

@MoisheKohan not more cumbersome and pedantic than having to define an ordered pair as $(a,b)=\{\{a\},\{a,b\}\}$, but here we are.

@ArturoMagidin this is the best answer I got so far, but I'm not sure about it yet. $(0,0)$ is a vector in the context of a specific vector space, but still I have not been given a formal definition of "vector". What is a vector? If it's an element of a vector space, then $(0,0)$ is a vector, but then the problems in the question come up: the fact that $(0,0)$ is a vector is given by the structure defined upon a set which $(0,0)$ is an element of. In a completely formal setting, you should specify such structure when speaking about $(0,0)$ as a vector. Hence the last sentence in the question.

How many math papers have you read (as percentage of all math papers which you have read) where the notation $\{x, \{x, y\}\}$ was used instead of $(x,y)$?

@MoisheKohan that's not the point. You still have that definition, which of course you don't mention every time you use an ordered pair. When you define what $(x,y)$ means though, you have to bring that up. It's the same thing when you define what "vector" means. After you have defined it, there's no reason to repeat its definition every time you mention it. But you have to define it first.

@Elvis: When one defines a vector spaces, one then states that the elements of the vector space are called "vectors" (or that vector space) and the elements of the field are called "scalars." That's the definition of "vector". When we say "Let $x$ be a vector",, we usually have a specific vector space (or an arbitrary but fixed vector space) in mind, which is understood from context. When it is not and there is a danger of ambiguity, we absolutely specify the vector space in question. (cont)

@Elvis: So when we say "$x$ is a vector", we are not making a formal statement in a formal language. We are using English to convey the information that $x$ is an element in a vector space structure which is either understood from context, or which can be fixed for the purposes of the sentence. You have not been given a formal definition of vector because there is zero point in having one, and it is useless to try to come up with one. Just like there is no legal definition of "registered student" without relation to a given class.

@Elvis When we say that the ordered pairs $(a,b)$ with $a,b\in\mathbb{R}$ are "vectors", it is implicit that we are talking about the vector space $\mathbb{R}^2$ with the usual structure. If we are not talking about that space, which is understood from context and long-standing convention, then we say so explicitly. Just like if I tell you that the real number $a\gt 0$ is a vector, if I don't say anything else you may assume I am talking about the usual vector space strcture of $\mathbb{R}$ over itself, and not something else. If I mean something else, I specify the vector space.

@ArturoMagidin oh, okay then. "The word vector doesn't have a formal definition" is the answer I was looking for. Thanks.

What is the point of your question? Your final sentence already shows you do know how to make precise the ungrounded naive vector definition in your first sentence, viz. a $V$-vector is any element of the vector space $V$, i.e. any element of its underlying set (universe). $\ \ $

@BillDubuque my question was whether my final sentence made sense.

Then you should explicitly ask that in the question.

@BillDubuque I explicitly asked "how is a vector defined?" and then wrote the definition I came up with. You could just answer the first question, which was explicit, and I would have had my second question answered implicitly, which is fine, since I asked it implicitly.

'I explicitly asked "how is a vector defined?"' I think if you look carefully, you'll find that you didn't. You have a bunch of statements with one apparently rhetorical question embedded in them. But you now have an explicit answer to the implicit question in your final paragraph.