Is the absolute value function a linear function?

A function $f(x)$ is linear if it satisfies the property $$f(ax+y) = af(x) + f(y).$$ Let's try $a=-1$, $x=1$, $y=0$: $$\begin{align*} |ax+y| = |-1| &= 1\\ -1\ |1|+|0| &= -1\end{align*},$$ so $f(x) = |x|$ is not linear.

Sometimes (especially in geometry) "linear" is understood to mean affine. A function $f(x)$ is affine if it satisfies the property $$f[ax + (1-a)y] = af(x) + (1-a)f(y).$$ Once again let's try $a=-1$, $x=1$, $y=0$: $$\begin{align*} |ax+(1-a)y| = |-1| &= 1\\ af(x)+(1-a)f(y) = -1\ |1| + 2\cdot 0 &= -1,\end{align*}$$ so $|x|$ isn't affine either.

Linear functions in analytic geometry are functions of the form $f(x)=a\cdot x+b$ for $a,b \in \mathbb{R}$.

Now try to write $\text{abs}(x)$ in such a form.

Another way to see it: linear functions are "straight lines" in the coordinate system (excluding vertical lines), this clearly excludes having a "sharp edge" in the graph of the function like $\text{abs}(x)$ has it for $x=0$.

In linear algebra (and this is the more common definition) linear functions denote ones of the form $f(x)=a\cdot x$ which is equivalent to require $b=0$ in the above definition. As $\text{abs}(x)$ is not linear with the first, weaker definition it cannot be linear either with this definition.