In calculus one calls $x \to ax + b$ a linear function. In linear algebra one calls $x \to ax + b$ an affine transformation, and says that it is linear only if $b = 0$.
The first order Taylor approximation to a differentiable map is reasonably called the linear approximation of the map. The terminology is reasonable because one is doing function theory, not linear algebra, and the meaning of the qualifier linear is context dependent. A formal way of giving content to the word "context" is to speak in terms of categories. Said more formally, what linear means when qualifying a morphism depends on the category to which the morphism pertains.
When working in the category of vector spaces (that is, doing linear algebra), a morphism is linear if it preserves the vector space structure, so $x \to ax + b$ is not linear, viewed as a map between one-dimensional vector spaces (it can be called affine if one works in the affine category).
When working in the category of smooth (to whatever degree) manifolds (that is, doing calculus), there is no notion of linear without some additional structure. A mapping ought to be called linear if it maps lines to lines, but for this to make sense there has to be a notion of line. The structure necessary to speak of lines is an affine connection - the lines are the geodesics of the connection. Properly speaking these are really lines only if the affine connection is torsion-free and flat (in this case there exist localy coordinates in which a geodesic really is a line). So, the additional structure that is used implicitly in speaking of a linear function is a flat torsion-free affine connection (on $\mathbb{R}$ it will be the usual derivative operator). A smooth mapping $f:M \to N$ between connected differentiable manifolds equipped with flat torsion-free affine connections can be defined to be linear if the covariant derivative of its differential vanishes (to make sense of this the differential $Df$ has to be viewed as a section of the bundle $T^{\ast}\otimes f^{\ast}TN$ equipped with the induced product connection; the condition is $\nabla Df = 0$). This is akin to defining a smooth map between connected differentiable manifolds to be constant if its differential vanishes, but is different in that it requires the extra structure of the flat connection (in fact the flatness is mostly irrelevant - dropping it one obtains the definition of a totally geodesic map - the flatness, which in the one-dimensional case is in any case automatic, is just there to justify using the word linear, because without flatness, there's no reason to think of geodesics as linear). If $M = N = \mathbb{R}$ with the usual flat connection, then this just means that $f^{\prime\prime} = 0$, so that $f$ has the form $f(x) = ax + b$.
