Essential norms of pointwise multipliers in the non-algebraic setting

Motivated by some recent results, but also referring to recognized classics, we compute the essential norm and the weak essential norm of multiplication operators acting between two distinct K{\" o}the spaces both defined over the same $\sigma$-finite measure space. A by-product of the technology we have developed here are some applications to Banach sequence spaces related to decreasing functions and to Banach spaces of analytic functions on the unit disc, in particular, Hardy spaces. We will close our work with some specific examples illustrating the previously obtained results including Musielak--Orlicz sequence spaces (in particular, Nakano sequence spaces) as well as Lorentz and Marcinkiewicz sequence spaces.