Hopefully I don't say something too stupid. I just wonder whether the definition of canonical might be relative.

For example, if we look at $ \mathbb Z / p$ as an additive group in fact there is no non-zero element which stands out. But if we look at $ \mathbb Z / p$ as a field $1$ stands out as a non-zero element.

Another example. From the geometrical reason alone, there is no good reason to choose a positive direction (Essentially there is no way to distinguish from left hand and right hand). But in a universe where there is electro magnetic force, we then have a canonical way to choose a positive direction.

Yet another example, there is a canonical way to choose whether you want a left shoe or a right shoe: If you are left-handed then choose the left one, if you are right handed choose the right one.

Perhaps what counts as canonical depends on where we are standing. A suggestion for a heuristic definition: canonical is definable with respect to the structure you are standing at.