Why is the chemical energy of a battery potential energy and not kinetic energy?

The energy is 'stored' in the battery. The energy in the battery is able to start a motion and by doing so transfering to kinetic energy, nut the chemical energy in the battery isn't 'stored' in movement.

Kinetic energy is simply the energy of something in motion,
and the battery is just sitting there, even if it's fully charged.

A fully charged battery has the potential to cause heating,
light, an electric current, or the motion of something like
a motor or the clapper of a bell.  What kind of energy it
produces depends on how you use it.  You might use it
to produce kinetic energy, and you might not.  Until you
connect it to something, its energy stays inside, in the
form of chemical energy, and nothing is moving yet.
 

Awoodchuck would chuck all the wood he could chuck if a woodchuck could chuck wood.