why aren't all lines of symmetry also diagonals?

The is wrong because when it says ALL, it means that EVERY symmetric line is a diagonal.

This can be proved wrong with a counterexample.

This here is a symmetric line, but not a diagonal

A diagonal is just a line from a vertex to any other vertex.

Try it with a normal peice of rectangular paper.

Fold it along the diagonal.

Does it fold exactly in half and do the 2 sides overlap each other perfectly? 

 If not, then that diagonal is NOT an axis of symmetry.