Is a 9 by 13 inch cake pan the same as a 13 by 9 inch cake pan?

None.

A centimetre (cm) is a measure of distance or length in 1-dimensional space. The cross-sectional of the cake pan is a measure of area in 2-dimensional space. The two measure different things and, according to the basic rules of dimensional analysis, conversion from one to the other is not valid.