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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^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 then Zariski neighborhood.

Similarly, is it possible to define tangent vectors for C^r-manifolds using ideals? Yes, simple take C^r-differentiable maps modulo 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 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.