It actually wasn't Hitler or Germany that first created the phrase.

It was Mussolini, in that he described the fate of history as running through the two nations, an Axis upon which time would turn into the new era.

Thus the Rome-Berlin Axis was the common name for the alliance between the states, which Japan became essentially a 3rd party to after trilateral negotiations.

The actual treaty they signed though was known as the Tripartite Pact. However several more powers did become party tot he treaty as they become the various and sundry "other nations" that fought on the Axis side like Romania and Hungary.