No, they thought of themselves as the Axis :P.

Axis didn't have the negative connotation it does today. It's unknown who exactly coined the term (likely the head of Hungary, but he died in 1936), but Benito Mussolini first used it in an official speech in 1936, describing the Italy-Germany friendship in the following language:

“This Berlin-Rome protocol is not a barrier, it is rather an axis around which all European States animated by a desire for peace may collaborate on troubles”

So, as you can see, he's using it in the mathematical sense. An axis is just something which is the center line of rotation. The idea was that Europe would revolve around this Rome-Berlin Axis.

Credit to a recent article by the Chicago Tribune here:

The case for Mussolini's claim is supported by reading all 16 articles leading up to his Milan address in The New York Times during October 1936. Reporters referred to an expected "accord," "pact" or "front" between Italy and Germany, but never "axis."

What we do know is that the Allies didn't officially start using the term to describe this friendship. Roosevelt's public papers did not mention "Axis" until November 11, 1940. It was a term already in use, but not officially among the allies yet.

This eventually formed into an actual alliance, called the "pact of steel," which never really caught on. After the Tripartite Pact (when Japan joined), the Axis term started picking up use worldwide. The terms of the tripartite pact essentially gave control of Europe to Germany and Italy, and Japan would be its own "axis" in Asia.