In Biberdorf, E. A. and Väth, M., On the spectrum of orthomorphisms and Barbashin operators, Z. Anal. Anwendungen 18, 1999(4), 12-31 it is shown that even in the more general case of an orthomorphism, the essential spectrum (for various definitions of “essential spectrum”) is the same as the spectrum and coincides with the “essential range” of the operator (for an appropriate definition of ”essential range”). Unsurprisingly, in case of a multiplication operator, the essential range of the operator is the essential range of the mulitplication function.
Note: To avoid a misunderstanding: This is a different sort of generalization than the original question in case $n=1$; the multiplication operator is not an orthomorphism in case $n>1$ (at least not with any order that I am aware of), except if it is diagonal a.e. (with the natural order) or, more genreal, diagonalizable a.e. (with respect to a measurable basis transformation). The latter is the case, in particular, if the multiplication operator is normal a.e.