Is Carter constant exclusive to general relativity? In other words, is there constant of motion analogous to Carter constant in any other field aside from general relativity?
I think since Carter constant is derived from Kerr metric - a metric describing rotating black hole  - and I'm not aware of any equivalent of Kerr metric in other fields of physics, this constant of motion appears only in general relativity. But I might be wrong.
 A: In other areas of physics the analogue of Carter's constant would be the concept of hidden symmetries and conservation laws corresponding to such hidden symmetries.
Remember, that Carter's constant arises due to the existence of rank-$2$ Killing tensor field ($K^{\mu\nu}$) in Kerr spacetime:
$$ C=K^{μν}u_μ u_ν.$$
If we write this quantity in terms of canonical variables and view it as a generator of infinitesimal canonical transformations, then such  transformations would leave Hamiltonian invariant, so it leaves the dynamics invariant. But because this constant of motion is of higher order in momenta the action of this symmetry on phase cannot be reduced to the lift of purely configuration space transformation. Such genuine phase space symmetries are called hidden, while the symmetries that could be obtained from configurations space transformations are called manifest. In the problem of particle moving in a Lorentzian metric manifest symmetries correspond to isometries of the metric and to Killing vector fields that generate such isometries. Hidden symmetries correspond to Killing tensors (and Killing–Yano tensors) of rank $≥2$.
And, of course, systems with hidden symmetries are not limited to general relativity. Examples of those include Kovalevskaya’s spinning top, Kepler problem, motion in the field of two Newtonian gravitational centers held fixed, quantum dots and particles with spin (examples from [1]).
For a review of the topic see the following paper:

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*Cariglia, M. (2014). Hidden symmetries of dynamics in classical and quantum physics. Reviews of Modern Physics, 86(4), 1283, doi:10.1103/RevModPhys.86.1283, arXiv:1411.1262.

A: In Newtonian gravity, you can construct a gravitational field that is somewhat analogous to Kerr in general relativity (it is stationary, axisymmetric, etc.) It turns out that for test particle motion in this field there also is an analogue of the Carter constant, making the motion integrable. (I’ll add some references later.)
Given the mathematical similarity between Newtonian gravity and electrostatics, I see no reason why one could construct an electrostatic analogue of this as well.
