What are exactly the various polarisations of the photon and how many are there? What are exactly the various polarisations of the photon and how many are there?
Are there:

*

*left and right : this makes $2$


*linear, circular, elliptic : this makes  $3$ (incompatible with the statement that there are two polarisations)
What is the intrinsic reason of two polarisations?
Is it due to the fact that there are two geometrical axes: $x$ and $y$?
Is it due to the fact that a photon is a vector? (spin 1)
 A: there are two polarizations (because $E\perp k$) and any other polarization is a linear combination of these two. The to main ones you can choose more or less freely. For example, if you choose the two orthogonal linear polarizations $(p,s)$ as the main ones, then the circular ones (right and left) are the sum of $(p,s)$ with a phase shift of $\pm \pi/2$. Then all elliptical polarizations can be obtained for other phase shifts.
A: The possible polarisations of light are due to the fact that light (and, indeed, any electromagnetic wave) propagates as a transverse wave in the electric and magnetic fields.
In an isotropic medium (including in a vacuum) a light wave can be expressed as the superposition of two plane waves which are perpendicular to one another. The waves have the same amplitude, but one is phase shifted with respect to the other.
With a  general phase difference, this produces elliptical polarisation. If the phase difference is $\pm90^o$ then this produces circular polarisation in one direction or the other. If the phase difference is either zero or $180^o$ then this produce linear polarisation. Both circular and linear polarisations are special cases of elliptical polarisation.
To specify the polarisation of a light wave (or any transverse wave) you therefore need two parameters. One defines the angle of the plane wave components (relative to a vertical or a horizontal plane, for example) and a second defines the phase difference between the plane wave components.
Note that this behaviour is classical behaviour that is common to all transverse waves. The quantum mechanical nature of light becomes apparent when we consider the polarisation states of individual photons.
A: As the title speaks of photons

What are exactly the various polarisations of the photon and how many are there?

The polarization of light is emergent from a superposition of photons, but photons are zero mass particles, and have either +1 spin or -1 to their direction of motion. The polarization of light is an emergent value, the  example for circular polarization:

A: Since the choice of a coordinate system is up to you, the polarization of a photon can be in any plane containing the direction of propagation.
