Unclear regarding Putin.

Stalin was probably the most powerful man on Earth during his lifetime, but even he didn't always get his way in the Politburo. Sometimes Stalin was on the losing side of a debate. But win or lose, the Soviet leadership emphasized speaking with a unified voice regardless of whether one had supported or opposed the outcome.

Is there a Politburo for Putin? Is there a group of people around him who can influence him and, sometimes, change his mind?

We don't know exactly, but there has to be a head of police, one of secret services, one for the military. All those people need to be trusted up to a point.