Why is Electrostatic Potential Energy positive when the charges are like?
There is no way to know the actual potential energy of a system, we can only determine it relative to some predefined zero value.  In the case of electrostatic potential, zero potential energy is defined as the energy between charges when their separation is infinite.

The charge of the electron doesn't change in this scenario; it is always $1.6\times 10^{-19}\ C$.

When oppositely charged particles near one another, they lose potential energy (their energy becomes more negative.)  When particles with the same charge near one another, they gain potential energy.  The attractive force between oppositely charged particles does work as their separation decreases, and the repulsive force between similarly charged particles requires work to be done on them in order to move closer to one another (so potential energy goes up.)