Forero, do have in mind something like "He thought they were Georgians, but then saw that they were flying an Azerbaijan flag"? I wouldn't be surprised to see that, but I would consider it an error for "Azerbaijani flag." "... flying the flag of Azerbaijan" would be fine. There isn't a "Bengal" to have a flag, but I would expect "Bangladeshi flag," not "Bangladesh flag."
In the U.S., we use "flag" without a possessive with "____ state": "At old Fort Niagara, they fly the United States flag, the New York state flag, the British Union flag of 1707, and the French Bourbon flag." But I certainly wouldn't write "the United States flag, the New York state flag, the Great Britain flag of 1707, and the Bourbon France flag." I don't know why.