I'm not a mathematician.
Merriam-Webster defines a spiral as "the path of a point in a plane moving around a central point while continuously receding from or approaching it."
Other sources I checked don't mention the word "continuously". MathWorld doesn't give a generic definition.
Is the Fibonacci spiral a "proper" spiral, given that the radius of curvature changes stepwise, and the centers of curvature are a set of individual points?
"Continuous" doesn't imply "at a constant rate", so the Fibonacci spiral certainly satisfies the dictionary definition.