“What the people in the street care about is how to get Daesh out of Iraq,” Ibrahim Bahr al-Ulum, a member of Iraq’s Parliament, said, using an Arabic name for the Islamic State, also known as ISIS or ISIL. “Now they feel Russia is more serious than the United States.”

As if to underscore that point, one widely viewed YouTube video shows Mr. Putin striding purposefully to the sounds of a patriotic Iraqi song, which hails him as a leader with the vision and determination to bring stability to Iraq.

“We don’t have to say his name; he knows himself well,” the singer belts out in the video, which ends with a clip of Mr. Putin conferring with President Hassan Rouhani of Iran.

Much of the popular fascination here with “Sheikh Putin” stems from the projection of sectarian politics onto the international stage.