First, it is not the "same charge". If you had the same charge on the two identical plates you would not have any potential difference between the plates.
The potential difference is due to the difference in the signs of the charges (one positive and the other negative).
The reason the charges do not move from one plate to the other is that the plates are separated by an insulator. If you connect the two plates with a metallic wire to create a conductive path the charge will flow and you may see a spark.
But for the charge to move through an insulator (or dielectric) you need a quite high electric field (see dielectric breakdown field). If you look at the label of a capacitor, you will see the maximum voltage indicated there.
As long as the voltage is lover than this nominal value, the charge cannot "break a path" through the insulator and the capacitor stays charged, as designed. But if you try to charge to 100 V a capacitor rated for 10 V is very likely that the charges will move from one plate to the other through the insulator.
In addition, when it breaks down the insulator (if it's a solid) may change its nature, chemically, like burning. So a permanent conductive path may appear.
If the insulator is a gas, you can have a gas discharge like in lightening.