In a particular mathematical context there may be an accepted format for describing an object. That way may be called the "canonical form". Usually it's established as a convention, not a formal rule.
In your examples you are talking about two different kinds of objects: linear operators and quadratic forms. Each has it's own "canonical form". They have nothing to do with each other.
The canonical form you refer to for linear operators has the adjective "Jordan" because it's one of many standard ways to write a matrix for an operator. Another "canonical form" is the matrix with respect to the usual (canonical) coordinate system (when that is relevant).