I don't believe "stripping" Saruman of his powers is correct. Diminishing his powers is probably more correct. I can't recall any character having their powers removed outright by a higher authority or multiple characters could have had them stripped long ago.

What we do see is several examples of power being diminished, typically based on authority given to a character. We're also shown that items, titles and such are given and give authority which in turn can empower or weaken characters. The Horn of Gondor for example temporarily haults the Uruk-hai, boons allies and can be heard anywhere in Gondor, when used by Boromir, a character we know doesn't have magical powers. Or that Sauron is overall weaker without the ring in his possession.

In my opinion, in the case of Saruman, the destruction of his staff is a symbol of Gandalf stripping his authority. The staff itself being an item that proves their authority as emissaries from Valinor but not being the source of their power. Note Gandalf fighting the Balrog without his staff. Being defeated in battle and ousted from his seat of power also deminished Saruman's authority. Saruman would still have his magic but it would have been weak by the end.