Does a AAA or D battery have an electric dipole moment? Why don't the opposite poles of two batteries attract each other like that of magnet's?
Not only is the capacitance and charge involved in a AA battery very small (I estimate 25 pF, assuming half-millimeter anode radius and half-centimeter cathode radius), but the inner structure is cylindrical:
This means that the stored energy in the battery is in an electric field which mostly points radially. If the field were purely radial, the dipole moment would be zero (though there would be quadrupole and higher moments).
Yes, in principle a battery does have an electric dipole moment.
The chemical reaction in a battery moves electrons from the cathode to the anode, leaving the cathode with a net positive charge and the anode with a net negative change. The charge will be given by the capacitance of the battery, $C$:
$$ Q = CV $$
where $V$ is the battery voltage i.e. about 1.5V. The trouble is that I don't know what the capacitance of an AAA or D battery is, but I would guess that it is vanishingly small. That means the charge separation and therefore the electric dipole will also be vanishingly small.
I would be very interested to know of experimental measurements of the dipole field of an open battery. Some Googling suggests the idea has occurred to other people but i wasn't able to find any trace of the experiment having been done.