The decision isn't his alone, there's A whole council. But in case he isn't happy with a Kage, he may gather a meeting and discuss the topic with others. His word of course means something, but isn't ultimate.
As I understand it, the Hokage is the equivalent of the top general or admiral of a military organisation who serves at the pleasure of the Daimyō and provide national security in return for certain benefits and privileges, e.g. holding lands and such. While the Daimyō ultimately decides who becomes Hokage (again, this is just from my understanding based on the manga), he refers to a comity of his top shinobi and delegates to advise him. For example, when Danzō become the Sixth Hokage Candidate, he was met with some opposition, and there was talk of Kakashi assuming the role of Hokage after Tsunade, however it was ultimately the Daimyō who gave the go-ahead.
To sum it up, the Daimyō wields considerable power, much like a King or a President in contemporary society.
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