The philosophy I subscribe to: **tangent vector** is an object defined by a **base point** and a **direction**.

When you want to make a formal definition out of it you do the following things:

* for **schemes**, you define tangent vector as a map from $\text{Spec}\\,\Bbbk[\epsilon]$;
* for **C<sup>r</sup>-manifolds**, you define tangent vector as a map from a neighborhood of 0 up to $\sim$

Is it possible to make a definition based on neighborhoods for schemes? Yes, you just have to do it technically right and consider *formal neighborhood* (which is the same for any smooth point of dimension n) rather then Zariski neighborhood.

Similarly, is it possible to define tangent vectors for C<sup>r</sup>-manifolds using ideals? Yes, simple take *C<sup>r</sup>-differentiable maps* modulo *C<sup>r</sup>-maps with 0 differential*. Those are proper analogues of $\mathfrak m$ and $\mathfrak m^2$ in this context. The second set, for a curve, can be described as set of *C<sup>r-1</sup>-maps multiplied by x*, but this is just one of the possible presentations.

**Conclusion:** when you look at it this way, the definitions are really similar on a deep level, but may be presented differently because of the technical details peculiar to each of the two cases.