What I mean by this question is, as we say that the electric field energy around a charge is the self energy of the charge, similarly can we say that the magnetic field energy around a current carrying wire is the Self Energy of the current? If yes, then following questions arise :

1. Why the current is associated with certain Self Energy?
2. Where does this magnetic field energy come from? Does it come from the battery the wire is connected to?
3. Why the self energy of a charge then called its interaction energy within and with other charges around?
4. Is the self energy of a charge equal to the integral of interaction energy over its volume? If yes, then what would be the Self Energy of an Electron?
5. Is there a difference between Self Energy and Potential Energy? Or they are just two names of the same basic underlying reality?
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It would be inappropriate to say that a current carrying wire would have Self Energy, in form of Magnetic Field Energy. Self Energy of charges are different from this concept. 

In electrical circuits, for a very small duration of time, when current was growing in the circuit, the current was variable, due to which the magnetic field associated with it was also variable, due to which induced non conservative electric field came into existence and this electric field, extracted some energy from the system (from battery) and stored this energy in form of magnetic field energy in the contour of circuit. We generally ignore this energy since this energy is very small, due to very small self inductance of a straight wire. Once the current becomes steady, no further storage of energy happens in magnetic field and magnetic field energy becomes steady. Now, all the further energy supplied by battery get consumed only in the resistance in joule heating. Now, if the circuit is broken or it is short circuited, then this stored magnetic field energy returns to the circuit and finally converts into joule heating.
Potential energy is a book-keeping device. Only *differences* in potential energy correspond to real energy. Since the electric and magnetic vector fields are defined by changes in potentials, any electromagnetic field with non-zero $\mathbf{E}$ or $\mathbf{B}$ has a (real) energy density, and this energy density is essentially $\mathbf{E}^2 + \mathbf{B}^2$ (ignoring some constants which I don't remember at the moment).

Self-energy can be defined as the energy of the electromagnetic field that surrounds an isolated charge, i.e. by integrating the field's energy density all over an otherwise empty space. If the charge is moving, part of that field energy comes from the magnetic field. However, a co-moving observer will only see a static electric field and a zero magnetic field, so the distinction between the electromagnetic field's electric and magnetic components is observer-dependent.

There is also a concept of self-energy in quantum field theory, which comes from a particle's interaction with itself. I don't really understand that stuff, so I'll just mention it here.