[The Squatter of Amon Rûdh]

I don't know a lot about the films, but depending on the context Saruman might think that Aragorn has the One, being that he is Isildur's heir and that Isildur was its last known owner. Saruman had devoted a great deal of time to the search for Isildur's body, and in Unfinished Tales we discover that he had found it, but not the Ring. Then again, being Saruman he might think that Aragorn would surely take the Ring from a Hobbit eventually.

Perhaps this scene was an attempt to explain Sauron's sudden interest in Minas Tirith, which in the book is the result of Aragorn looking into the Palantir of Orthanc, but not having watched the extended editions the above are all guesses.

Saruman knew a great deal about ring-lore; in fact he claimed to have made a ring of power for himself, although what powers it conferred, if any, were never disclosed.

As far as I know, the Ring of Barahir had no power other than as a token of the alliance between Finrod and its bearer.