Your operation is the multiplication in the algebra of functions $\{1,\ldots,n\}\to\mathbb R$.
This algebra is a vector space with the obvious pointwise addition and scalar multiplication. It is isomorphic as a vector space to the usual $n$-dimensional vector space $\mathbb R^n$ (depending on the details of the formalization you work in, they may even be identical rather than simply isomorphic). Nevertheless, it is most common to think of it as something separate from $\mathbb R^n$.
Why is that? It is because usually when we speak about $\mathbb R^n$ we want to think of it as a concrete representation of some other vector space which we get by choosing a basis, and in most cases we have considerable liberty to choose the basis. Therefore $\mathbb R^n$ usually invokes an idea of "some abstract $n$-dimensional vector space with a more or less arbitrarily chosen basis".
However, your multiplication operation does not fit into this framework, because it is not preserved when we move to another basis. Therefore it is not a useful concept to speak of in the common situation where the basis was chosen more or less arbitrarily.
So in order to reduce the risk of confusion, we usually pretend that the component-wise product doesn't exist when we speak merely of $\mathbb R^n$. In the few situations where the multiplication is useful, we prefer to call the space something else such as to remind ourselves that our usual freedom to switch to another basis whenever it's convenient doesn't exist anymore.
In particular, you won't find the the Hadamard product listed in an article about "Euclidean vectors" -- because what makes the vectors Euclidean is that the relations between them do not change if we rotate the coordinate system. But the product does change if the coordinate system is turned.
Saying that a vector space is a set whose elements can be added etc. can be seen as a remnant of an "Eulerian" worldview. In this view, objects such as numbers have an inherent concept of arithmetic operations such as addition and multiplication. $2 + 3 = 5$ no matter whether you are considering the integers or the complex numbers, and they are the same $2$, $3$ and $5$. In this worldview, being a vector space is a property. A set is either a vector space, or it's not.
Modern mathematics tend to take an alternative worldview. Here, the elements of a set has no internal structure, $2 \in \mathbb R$ is merely a label for an element, and switching labels around doesn't affect mathematics whatsoever. To meaningfully talk about addition on $\mathbb R$, you need to equip it with additional structure, i.e. the structure of a function $\textsf{add} : \mathbb R^2 \to \mathbb R$. In this worldview, a set can become a vector space in multiple ways.
So saying a vector space $\mathcal V$ "is" a set doesn't make sense, in the say way that saying a two dimensional point $(x,y)$ "is" the real number $x$ doesn't make sense. Since there may be different points $(x,y_1), (x, y_2)$ that have the same $x$.
Similarly, an inner product space $\mathcal V$ isn't a vector space, and a metric space isn't a topological space.
To contrast this, it does make sense to say a finite-dimensional vector space $\mathcal V$ is a vector space, since being finite dimensional is a property. And a metrizable space is a topological space.
This distinction of stuff, structure and property has already occurred to many mathematicians, and it is articulated and promoted by John Baez, Michael Shulman and others.
With this in mind, it's easy to see why saying "a vector is an element of a vector space" is a correct but less-than-useful statement. Given a mathematical object $v$, it is no inherent property of $v$ that $v$ belongs to some vector space. You can construct any set $V$ with $v$ in it, and $v$ will become a vector in $V$. What's more useful is saying "$v$ can be viewed as a vector in the vector space $V$". This is telling the reader to think about $v$ in a different and possibly fruitful way.
Of course, usually when people say "Let $V$ be a vector space", what they usually mean is "Let $(V, +, \times, \dots)$ be a vector space", and the letter $V$ actually refers to the set instead of the vector space. So notations such as $x \in V$ or $W \subset V$ makes sense because they are really just notations of sets. But this is just a matter of trivial notation fiddling, and doesn't bear much value unless there is danger of confusion.
Best Answer
A vector space is any space where you can do addition and scaling in a way that "makes sense". It does not nessesarily have to align with your intuition of containing "arrows" that point somewhere in physical space, that is just starting intuition for it.
The formal definition of a (real) vector space is that it's a set $V$, containing a special element $0$, whose elements can be added together and scaled by real numbers. It also has to follow the basic algebraic rules you'd expect it to follow. Formally, we have for all $r_1,r_2 \in \mathbb{R}$ and $v_1,v_2,v_3 \in V$,
$$(v_1 + v_2) + v_3 = v_1 + (v_2 + v_3),$$ $$0 + v_1 = v_1,$$ $$v_1 + (-1)\cdot v_1 = 0,$$ $$v_1 + v_2 = v_2 + v_1$$
And
$$r_1(r_2\cdot v_1) = (r_1r_2)\cdot v_1,$$ $$(r_1 + r_2)\cdot v_1 = r_1\cdot v_1 + r_2 \cdot v_1,$$ $$r_1\cdot (v_1 + v_2) = r_1\cdot v_1 + r_1 \cdot v_2.$$
Any set equipped with these operations that follows these rules is a vector space, and its elements are definitionally vectors.