Your character’s travel documents would have been Ottoman (like those of my grandfather, who left Beirut for Brazil around 1905). The Mount Lebanon Mutasarrifate was an administrative unit from 1860 to the dissolution of the Ottoman Empire in 1918. It covered only part of the area of modern Lebanon, but if your character came from a town in that area, it would be reflected in their travel documents. There were also the sanjak of Beirut and Sidon, and Beirut vilayet. Sanjak and vilayet were administrative layers in the Ottoman provincial organization. Lebanese did not love the Empire. A person from anywhere in the area could have identified themself as Lebanese, or by their religion. The Druze and Maronite faiths in particular are almost synonymous with Lebanese geographic origin. The character might also refer to their hometown, Baalbek or Baakline or wherever.

Here’s a paper about Ottoman travel documents and internal controls on movement.

Here’s a good book on the development of Lebanese identity under the Ottomans.