Equivalence?

From the book Not Even Wrong Dr. Woit makes a common mistake on page 6 explaining Einstein's famous equation E=mcc:

Note that using units in which the speed of light [c] is set to one simplifies this [equation] to E=m, so energy [E] and mass [m] become equal in the context described by this equation.  As a result, particle physicists use the same units to measure energy and mass.

If this was truly the case then the equation should more properly be E=xm where x could be any real number without any units of measurement.  There is no context in which energy and mass are equivalent.  They are measured in different units which cannot be ignored in any context. For example, in the equation for area of a rectangle (A) with length r and width w given by A = rw, if the length is set to 1 unit in size, A = w does not imply that area and width are now equal in the context described by this second equation.  Area is still given in squared units of displacement while lengths and widths are stated in units of displacement that are not squared.