The archetypical example of a mechanical system is a particle propagating on a manifold . The phase space of this particular system happens to be canonically identified with the cotangent bundle of . Here the covector in over a point is physically interpretd as describing a state of the system where the particle is at position and has momentum (essentially: speed) as given by .
Therefore locally for coordinate patch the -canonical coordinates of the Cartesian space are naturally thought of as decomposed into “canonical coordinates” on the first factors and a set of “canonical momenta”, being the canonical coordinates on the second -factor.
Notice that “canonical” here refers (at best) to the canonical coordinates of the Cartesian space once has been chosen. The choice of however is arbitrary. Hence, despite the (standard) term, there is nothing much canonical about these “canonical coordinates” and “canonical momenta”.
Therefore generally, in the context of mechanics, with such a local identification one calls the canonical momentum of the coordinate (or sometimes “canonical coordinate”) .
Globally the notion of canonical momenta may not exist at all. The notion that does exist globally is that of a polarization of a symplectic manifold. See there for more details.
Discussion of how there is a flat connection on the bundle of spaces of quantum states over the space of choices of polarizations of a symplectic vector space and how this reproduces the traditional relation between canonical coordinates and canonical momenta by Fourier transformation? is in (Kirwin-Wu 04).