In linear differential equations
models are said to be linear iff superposition holds, f(x+y) = f(x) + f(y) and cf(x) = f(cx)
but what about a single linear variable equation?
f(x) = m * x + h,
f(x+y) != f(x) + f(y)
cf(x) != f(cx), because the h it will have a bias.
A simple linear relation doesn't satisfy superposition.
Here linear is used in two different ways. The function $$f(x)=ax+b$$ is linear in the sense that it is a polynomial function of degree $1$. As you have noted, though, it doesn't exhibit linearity. This function is only linear in the additive sense if $b=0$. When $b\neq 0$, it is a linear function composed with a translation, which is called an affine function.