What is the analog to the electric field for those particles (different to photon) that have a polarisation? Photon has a polarisation. The polarisation defines how the two components of the electric field evolves with times in the transverse plane.
Those particles, different to photon, that have a spin $s$ have several possible polarisations: $2s+1$.
What is the analog to the electric field (for the photon) for those particles (different to the photon) that have a polarisation? Indeed, left and right, for photons, are defined for the electric field behaviour. If there is no analog for other particles of the electric field, I don't see any definition.
 A: The formula $2s+1$ refers to massive particles, which mediate short-range forces, unlike electromagnetism. Massless particles have only two polarization states, for any spin.
So, the analog to the two polarization states of the electromagnetic field, for higher spin fields, is still two polarizations. It is the exact same thing.
For massive bosons like $Z,W^\pm$, or confined bosons like gluons, there is no macroscopic force, so no classical analogue. But if given one classical field, with non-trivial effects over macroscopic distances, then the different polarization states are still referred to as "polarization states", and they have the same physical interpretation, namely they describe the different ways the field may oscillate as it propagates. For example, a massive vector field would have the two standard polarizations, describing orthogonal oscillations, plus the extra (massive) mode, describing longitudinal oscillations.
A: Electromagnetic waves, hence photons, are described by a vector field$^*$. A vector field has 3 (2S+1) components and corresponds to spin S=1. In the case of electromagnetism only the 2 spin components parallel and antiparallel to propagation are allowed because of charge conservation. The direction of this vector field can rotate about the direction of propagation. Its time derivative is the electric field associated with the wave.
Particles with different spin values are described by different types of field. For example protons, electrons and positrons are described by Pauli spinors$^{**}$. These have two components corresponding to S=1/2. The components of such a spinor are the analogues of the components of the electromagnetic vector potential that you are asking for. It is not generally so that the time derivative of these components are interpreted as some kind of force, as in the electromagnetic case. Spinors are conceptually more difficult than vectors. See https://en.wikipedia.org/wiki/Spinors_in_three_dimensions
There are also elementary particles with spin S=1, for example the pions or even S=3/2. These are mesons made up of two or baryons made up of 3 quarks (each s=1/2).
$^*$ Coulomb gauge assumed.
$^{**}$ Non-relativistic approximation. Dirac spinors are used in the fully relativistic approach.
A: From the solid state point of view one can think of many quasiparticles analogous to photons, and described formally in a similar way, which would have something else in place of electric field.
The most obvious example is phonons, which are essentially quantized elastic vibrations of the lattice, and which have three polarizations: two transverse and one longitudinal, which are the directions of the lattice vibrations in respect to the wave vector.
The other boson-like excitations, such as magnons, plasmons, spinons, all have as polarization the change of the underlying physical field.
Disclaimer: this perhaps takes away from the direction that the OP had in mind, but necessary for seeing the broader picture, in my opinion.
