Consider a simple switch.
It's very linear to the signals which it switches
It's very non-linear from the position of the toggle to the effect on the signals which are applied to the contacts.
Semiconductor manufacturers have never had life easy deciding whether to put mixers in their linear or the non-linear databooks
Of course a pure analogue multiplier is a linear function, but look inside at the non-linear functions needed to implement it! Usually a true Gilbert Cell.
David