Why do we think of a vector as being the same as a differential operator?
Solution 1:
There are three common ways to define tangent vectors $v\in T_{p}M$ on abstract manifolds:
As linear operators $C^{\infty}\left(M\right)\to\mathbb{R}$ satisfying the Leibniz law $v\left(fg\right)=f\left(p\right)vg+g\left(p\right)vf$
As equivalence classes of curves satisfying $\gamma\left(0\right)=p$, where two curves are equivalent if their first derivatives at zero agree in some chart
As assignments of tuples $v_{\varphi}=\left(v_{\varphi}^{1},\ldots,v_{\varphi}^{n}\right) : \operatorname{dom}( \varphi )\to \mathbb{R}^{n}$ to charts $\varphi$ such that $v_{\varphi},v_{\psi}$ are related by the Jacobian of $\varphi\circ\psi^{-1}$.
The reason the first is so popular is that it does not require one to talk about coordinate charts - all the dependence on coordinates is encapsulated in the dependence on the ring of smooth functions, $C^{\infty}\left(M\right)$. Once one has established the well-definedness of $C^{\infty}\left(M\right)$, this definition is truly coordinate-free and self-evidently well defined. This "cleanliness" is just an aesthetic advantage (one can of course check that the other definitions are perfectly consistent and that all three are equivalent), but it's a significant one - it's much easier to handle the concepts in DG if you keep the definitions as clean as possible, even if computations sometimes require one to fix a chart and get their hands dirty.
Now, as to interpretation - I argue that in general, the first and second definitions above are much more natural interpretations of what a vector actually is than the third. My reasoning stems from thinking about what you can actually do with a vector in an abstract smooth manifold.
In an affine space, the natural interpretation of a tangent vector is as a displacement - one can take a vector $v$ at a point $p$ and translate over to $p+v$; and this addition is literally addition of components in cartesian coordinates, so we have a very close relationship with the vector components.
This does not work on a general smooth manifold - you need at least a metric for this to have any analog at all (the exponential map), and even then it's not true that $``p+v"^i=p^{i}+v^{i}$. In the abstract case, vectors indicate an "infinitesimal" direction, which can be formalised as one of the first two definitions I gave above:
vectors are the directions in which you can differentiate functions; or
vectors are the directions in which curves can be travelling.
The picture you have in your head should look the same as before - vectors are still directions with magnitudes attached to points. The difference is that since there is no longer a literal formalisation of this concept (displacements in affine spaces), we have to choose a new formal definition of what a "direction" is; and I believe that while the component definition is the most familiar in terms of computation, it is not a good conceptual model.
Solution 2:
In this paper by David Hestenes, a modern proponent of applying clifford algebra in physics, Hestenes explores some of his own frustration with this identification. He attributes the "formula"
$$e_i = \frac{\partial x}{\partial x^i}$$
to Cartan. Hestenes concludes that Cartan did not mean this literally (when you write the derivative as a limit, a difference between points has no meaning) and merely used it as a "heuristic device." Since none of Cartan's later arguments relied on this particular point of logic, it didn't matter. Hestenes attributes it to later mathematicians that the subtle nonsense in the formula involved, useful though it may have been, drove people to abandon it by dropping the $x$, leaving
$$e_i = \frac{\partial}{\partial x^i}$$
and thus the identification of tangent vectors and partial derivatives was born. Is this accurate to history? Not sure. But it makes a nice and concise story to tell your children who are struggling with their differential geometry bedtime stories.
Anyway, Hestenes has reasons for attacking this identification. It's fundamentally incompatible with the greater structure of clifford algebra (partial derivatives commute, whereas clifford algebra demands a wedge product of vectors that anticommutes).
However, clifford algebra as applied to differential geometry can, in principle, neatly dodge some of the problems involved in having to consider a proliferation of charts to get anything useful done. By considering an isomorphism between a typical manifold and a so-called "vector manifold," Hestenes turns the points on a manifold into vectors in some vector space (in other words, he embeds everything; the arbitrary nature of embedding is dodged by embedding in an infinite dimensional vector space) and is able to identify $x$ as a vector and thus $\partial x/\partial x^i$ as a meaningful geometric quantity.
At any rate, what I wish to point out is that while the identification of tangent vectors and partial derivatives may have had good motivations, it does deprive one of being able to apply clifford algebra to differential geometry problems.
Given the power of clifford algebra in concisely describing geometric quantities beyond simple vectors, this is a nontrivial price to pay, in my opinion. Being able to algebraically handle the tangent space not as some abstract thing that holds vectors but as an oriented $k$-vector called the pseudoscalar of the manifold is particularly powerful, for instance. It makes questions of orientable manifolds trivial because the pseudoscalar is multi-valued for nonorientable manifolds. And when phrased that way, orientability becomes one of the most natural things in the world. Compared to the clumsy, clumsy definitions you usually see in differential geometry, involving determinants of transition maps and so on, a clifford algebra viewpoint of differential geometry puts much more emphasis on the geometry, in my opinion.