It is not an answer but it must have to do with the way our brain pick up a sample between a few ones. It must be the  minimum of some function which can be implemented for real in the brain. I do not believe that there is a pure logical definition of "canonical" independently of the way our brain works. Experience : give me a number? What do you answer? 0 or 1 rarely $\pi$ or even 115674. The numbers 0 and 1 are canonical in some sense. Give me a basis of ${\bf R}^3$. The same holds $((1,0,0),(0,1,0),(0,0,1))$ I minimize the number of different digits and I pick them in my "basis" of canonical numbers. Well, interesting question.

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What is the canonical circle ? Ce circle in ${\bf R}^2$, centered at $(0,0)$ with radius $1$. 

I know two numbers $0$ and $1$, the radius cannot be $0$ because it is not a (true) circle, so the radius is $1$, now the center could be $(0,0)$,   $(0,1)$,  $(1,0)$ or  $(1,1)$ ? I prefer $(0,0)$, $0$ is simpler than $1$. How do you fit this example with category arguments?

<i> BTW I have nothing against category theory, I like it. But I'm curious to see if this example fits general categorical arguments.</i>