It's not nearly as mystical as folks like to make it sound. A lot of the language around it dresses it up to appear more than it is in an effort to be rigorous.
Systems that are both linear and time-invariant, sometimes abbreviated to LTI systems, are the most common sort of thing (i.e., model) that one actually analyzes. When time-dependence is present, it is always modelled explicitly and usually modelled as a complication of a time-invariant system.
Not sure where you were taught that, but it does not strike me as the generally accepted meaning of the term "linear".
Where I hail from, linear means:
f(a*x + b*y) = a*f(x) + b*f(y)