Why the capacitor stores energy but not charge??

First of all, let's know what really storing anything means. To store something implies there should be presence of some extra amount of it in them. Like same storage of energy or charge in any system implies the net charge or energy in the system is not equal to zero.

Now let's consider a parallel plate capacitor in which the plate  connected to the positive terminal of a battery gets positive charge and induces same amount of negative charge on the another plate. So the total net charge becomes zero and hence the capacitor does not store charge and hence its plates only do the separation of charge.

Now let's talk about energy. As there is positive charge on the first plate so it will produce field lines that will create an electric field between the both plates and that cause storage of energy. When we connect the plates generate work and this phenomenon become possible only because the capacitor has stored energy.

So, in the nutshell as the capacitor has net charge zero it doesn't store any kind of charge on it but meanwhile whenever charges of opposite polarity are separated then electrical energy is stored within the capacitor by the same charge present within it. So a capacitor stores energy but not charge.

However, it does not implies that the capacitor does not have any charges, it has a net charge of zero.