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 $\text C^r$-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 than Zariski neighborhood.
Similarly, is it possible to define tangent vectors for $\text C^r$-manifolds using ideals? Yes, simply take $C^r$-differentiable maps modulo $\text C^r$-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 $\text C^{r-1}$-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.
