They may believe the same thing, at least religiously, but the Saudis don’t won’t a nuclear power next-door, especially run by fanatics. Saudi Arabia is a strong and staunch supporter of Islam. But it is like the old joke about certain aspects of Christianity, I believe in the afterlife, I’m just not ready to go today ... or something like that.
Still don’t trust them. And when Iran is eliminated, they’re in the power seat, both economically AND religiously.
Add to that the Iranians consider the House of Saud to be something akin to heretics.
Add to that the whole Sunni vs. Shia thing on top of the ethnic rivalries between the Persians and the Arabs.
The Saudi royal family wants to toe the line between Islamic and Westernized. It’s good for business.