I think you'll find that derivatives are overwhelming defined as operators on vector spaces. It is definitely correct to talk about a linear function on a vector space.
I agree with you, but this is not what GGP is saying. GGP is saying another true fact (i.e., that derivative of a linear function is that same function), which a different thing than stating, as the article says, that the differentiation operator is linear. On a manifold there is no concept of a linear function, so you cannot say that the derivative of a linear function is the same function, but the differentiation operator is still defined and linear.
Of course you are correct, even though one could argue the linearity of differentiation is a property you can obtain by differentiating in coordinate charts, where the reasoning is still valid.
In any case, it is probably a good thing to get a good intuition of what differentiation and derivatives are in the vector space setting before digging into differential geometry.