According to this Wikipedia site https://en.wikipedia.org/wiki/Tangent_space
"We can therefore define a tangent vector as an equivalence class of curves passing through x while being tangent to each other at x."
and
"... equivalence classes of such curves are known as tangent vectors of M at x."
However, the figure below shows a tangent vector v on TxM, a plane off M. How can this vector v, a straight line off M, be an equivalence class of curves on M?
$\newcommand{\Reals}{\mathbf{R}}$The blue vector in Wikipedia's image is a representation of the common velocity at the point $x$ to an equivalence class of curves in $M$. To explain the sense in which the blue arrow "represents a velocity" we have to be honest about a certain "abuse of geometry" inherent such a diagram.
Cartesian three-space $\Reals^{3}$ has a tangent space $T_{x}\Reals^{3}$ at each point $x$, consisting of all velocities $\gamma'(0)$ of differentiable paths satisfying $\gamma(0) = x$. This tangent space is more-or-less naturally identified with $\Reals^{3}$ itself: We view each vector $v$ as the velocity at $t = 0$ of the path $\gamma(t) = x + tv$, or using Wikipedia's definition, with the set of differentiable paths through $x$ and having velocity $v$ at $x$. However, the manifold $\Reals^{3}$ and the tangent space $T_{x}\Reals^{3}$ are not literally the same entity.
The last point is clarified by considering the family of all tangent spaces, (the total space of) the tangent bundle $T\Reals^{3}$, which we identify with $\Reals^{3} \times \Reals^{3}$. An ordered pair $(x, v)$ may be viewed as a tangent vector at $x$. The vector space structure (vector addition, scalar multiplication) resides only in the second component: For an arbitrary point $x$, for vectors $v$ and $w$, and for a scalar $c$, we have $(x, v) + c(x, w) = (x, v + cw)$. This is the only meaningful vector operation on pairs $(x, v)$. The manifold $\Reals^{3}$ is identified with the set of $(x, 0)$, a.k.a. the zero section of the tangent bundle. The tangent space $T_{x}\Reals^{3}$ is the set of pairs $(x, v)$ with $v$ in $\Reals^{3}$.
We're now in a position to explain the abuse of geometry mentioned earlier: The tangent bundle of $\Reals^{3}$ is six dimensional, and therefore vexing to visualize. We may, however, depict a tangent vector $(x, v)$ as the arrow in $\Reals^{3}$ from $x$ to $x + v$. In other words, we map the tangent bundle to Cartesian space by the mapping $(x, v) \mapsto x + v$. The velocity $(x, v)$, which lives in the tangent bundle, is now represented in the same Cartesian space containing the objects (typically curves or surfaces) that gave rise to the velocity we wanted to picture. Similarly, the "tangent plane" $T_{x}M$ in the diagram is a subspace of $T_{x}\Reals^{3}$; our conventional representation maps this plane to the plane through $x$ that "looks tangent" to the surface $M$. The vector space structure is defined geometrically by arrows based at the point of tangency. Strictly, this plane does not occupy the same space as $M$, either.
Moving beyond the scope of the question (as it were), if we tried to draw two tangent planes to $M$ at different points, the representations would generally intersect along a line, but the tangent spaces themselves do not intersect, because they are subsets of distinct tangent spaces of $\Reals^{3}$, and distinct tangnt spaces are disjoint.
The modern viewpoint of abstract manifolds defines tangent vectors as "data structures" generalizing the "classical" picture in Cartesian space, but in a way depending only on the manifold structure. (Particularly, geometers prefer not to think of manifolds as subsets of a Cartesian space, since that fixes additional structure.) "Equivalence classes of velocities of differentiable paths" is one idiom. But there are other definitions, such as derivations (first-order differential operators) on the algebra of smooth functions. The crucial feature, whatever the definition, is that tangent vectors transform compatibly with the chain rule under change of coordinates.
The figure in Wikipedia has only motivational purposes. It depicts a two-dimensional smooth submanifold of $\mathbb R^3$.
For smooth submanifolds $M \subset \mathbb R^n$ one can define a geometric variant $\tilde T_xM$ of the tangent space at $x \in M$ as follows:
Consider a smooth curve in $M$ through $x$, i.e. a smooth map $u: J \to \mathbb R^n$, where $J \subset \mathbb R$ is open with $0 \in J$, such that $u(J) \subset M$ and $u(0) = x$. The tangent vector of $u$ at $t = 0$ is the (standard) derivative $u'(0) \in \mathbb R^n$. The Euclidean tangent space $\tilde T_xM$ is the set of all such $u'(0)$ where $u$ is a smooth curve in $M$ through $x$. In the Wikipedia-figure the blue vector is such a tangent vector. In fact, $\tilde T_xM$ is not contained in $M$ in this example, but it is what we intuitively understand as the tangent plane of $M$ at $x$.
In an abstract manifold $M$ (which is no submanifold of some $\mathbb R^n$) we can still consider smooth curves $u$ in $M$ through $x$, but there is no geometric definition of $u'(0)$ as above. So what can be done? Going back to a smooth submanifold $M \subset \mathbb R^n$, we see that tangent vectors in $\tilde T_xM$ are in $1$-$1$-correspondence with equivalence classes $[u]$ of curves $u$ in $M$ through $x$, where $u \sim v$ if $u'(0) = v'(0)$. This observation can be used to define tangent vectors in any manifold $M$ as equivalence classes of smooth curves $u$ in $M$ through $x$. Details can be found in The motivation for a tangent space.
Update:
The OP correctly comments that for a submanifold $M \subset \mathbb R^n$ a geometric tangent vector $u'(0) \in \mathbb R^n$ at $x \in M$ is not identical with an equivalence class $[u]$ of curves $u$ in $M$ through $x$ having the same derivative at $t = 0$. There only exists a $1$-$1$-correspondence between these two types of objects.
It is, however, a philosophical question what a tangent vector really is. There are various technically different approaches to introduce this concept, but they are all equivalent.
For a submanifold $M \subset \mathbb R^n$ we can define the geometric tangent space $\tilde T_xM$ as above. It turns out that there is a canonical isomorphism to the variant of the tangent space $T_xM$ based on equivalence classes of curves. Intuitively such an equivalence class of curves specifies a velocity vector on $M$ at the point $x$: All particles moving along any curve in a fixed equivalence class have the same velocity (direction + absolute value) when they pass through $x$. If the force keeping the particle in its path $u$ through $M$ would vanish at $t = 0$, then the particle would move further along the straight line going through $x$ in direction $u'(0)$.
On abstract manifolds $M$ it is impossible to define a definite "numerical" value $u'(0)$ of a curve $u$ in $M$ in through $x$, but we still can use the "equivalence classes of curves" definition. Thus we may regard this approach as more general, but perhaps less intuitive.
A third approach on an abstract $M$ is to define a tangent vector at $x \in M$ as a derivation at $x$ on the algebra $C^\infty(M)$ of smooth functons $f :M \to \mathbb R$. See your Wikipedia article. A derivation can be regarded as a generalized directional derivative as known from multivariable calculus. This approach is even more abstract, but it results again in an isomorphic tangent space.
All approaches have their benefits, and it would be inadequate to claim that one of them is the sole legitimate variant.
There are similar phenomena in many branches of mathematicas. For example, when we construct the real numbers as an extension field of the rational numbers, we can do it via Dedekind cuts or via equivalence classes of nested rational intervals or via equivalence classes of rational Cauchy sequences. The results are technically different, but all are isomorphic. So what is a real number? In my opinion this is not really important, the real numbers are characterized by their properties which do not depend on the specific construction.
Here are some related questions:
Alternative concepts for tangent spaces of smooth manifolds and derivatives of smooth maps
The point is that they're trying to introduce the idea of defining a tangent vector, a think you'd normally think of as sticking out from the manifold, as an object on the manifold.
In other words, $v$ at $x$ is the thing you'd naively think of, but we're trying to think in terms of $\gamma$, a curve on the manifold passing through $x$.
More generally, a big theme in the elementary theory of manifolds is trying to get away from having to think of your manifold as being in some bigger space. This is just part of that.
Also keep in mind that there are multiple ways to define tangent vectors, so this curve-based method is just one.
