I would say that "canonical" ought to be used to describe when no choices have been made.

A nice example of a non-canonical identification: A principal bundle is made up of principal homogeneous spaces for the action of a Lie group. These are spaces which are homeomorphic but non-canonically isomorphic to the Lie group. For example, I might have a circle bundle. My Lie group would be a `concrete version' of the group such as $\{|z| = 1\}$, but my fibres are simply circles. I would need to *choose* a base point on each of the circles to make them into groups in the same way. This amounts to taking a global section and can't always be done (e.g. circle bundle on the sphere has no global section by hairy ball theorem), so the non-canonical-ness might actually be important

The labelling of identifications as Canonical and Non-canonical is common in linear algebra: Since one chooses bases so often, it is worth pointing out when such a choice is avoided...

To prove that $V^* \otimes V^*$ is isomorphic to $(V \otimes V)^*$, one ought to work with elements of the spaces directly rather than their representations in some basis. I would therefore call the resulting isomorphism `canonical'.