Why is induced electric field (due to a changing magnetic field) in the form of concentric circles? What can't it be any other shape, like concentric squares, or rectangles, etc? Also, where is the common centre of the electric field lines?
 A: The field lines formed by a changing magnetic field are not necessarily circles;  all that is necessary is that they obey $\nabla \times \mathbf{E} = - \partial \mathbf{B}/\partial t$.
If it happens that we have cylindrical symmetry in the system (as is the case in, say, the changing electric field inside a solenoid, or between two circular poles of an electromagnet), and if there is no charge present, then it is possible to argue from symmetry that the field lines are circles.  Specifically:

*

*The electric field vector is independent of angle and position along the axis, by symmetry.

*The electric field vector can't point radially inward or outward.  If it did, we could draw a cylindrical Gaussian surface with a net electric flux through it;  but we are assuming that no charge is present, so this would contradict Gauss's Law.

*The electric field can't point along the axis by reflection symmetry.

Combining these facts, the electric field therefore must be of the form $\mathbf{E} = E(r) \hat{\phi}$, which implies circular field lines that are coaxial with the axis of rotational symmetry.
However, if we don't have symmetry of this type, then all bets are off.  In fact, situations in which the field lines of a divergence-free vector field (such as $\mathbf{E}$ in the absence of charge, or $\mathbf{B}$ in general) form closed curves are generally quite rare.  There's a nice old article from 1954 that discusses this further in the context of magnetic field lines, but much of that discussion can be applied to the topology of electric field lines in the presence of steadily changing magnetic fields:

K. L. McDonald, "Topology of steady current magnetic fields", American Journal of Physics 22, 586 (1954).

