This is clearly kidnapping. It is probably not a terribly aggravated sub-type of kidnapping, but it is kidnapping nonetheless. It is probably a felony.
The fact that the victim does not press charges, or ratifies the conduct after the fact, does not change the fact that a crime was committed.
The police decision to arrest the ex-boyfriend was entirely proper. It was not a false arrest. They had probable cause to believe that a crime was committed by the ex-boyfriend, and, in fact, a crime actually was committed by the ex-boyfriend.
In the United States, the prosecutor has full authority to prosecute the ex-boyfriend to the full extent of the law for felony kidnapping for his conduct, over the objections of the victim. The decision to prosecute or not is entirely in the discretion of the prosecutor who brings criminal charges on behalf of the state and not the victim. Often the police and prosecutors will honor a victim's wishes, and this appears to be what happened in this case, but they are not required to do so, and could change their minds and bring charges in the future against the ex-boyfriend within the statute of limitations, if they wanted to do so.
(This analysis does not apply in countries with Islamic law, but the facts of the question suggest that Islamic law does not apply in this jurisdiction.)