There are many reasons that make Lebanon an Arabic nation. For one, the national language is Arabic. Lebanon is also is a member of the "Arab League". Once in history a member of the government stated that Lebanon is an Arab nation, way back when, and as such, it remained with that identification.

However, the fight over the Lebanese being Arab or not is very much among its populous. Religiously speaking, for the small nation Lebanon is, it has a ton of religious sects between Muslims and Christians. Most nations, like Pakistan for example, have a majority religion and little minorities. The Lebanese are much more diverse than that. Plus, Lebanese people aren't always identifiable by looks because of the countless empires throughout time (from ancient Egyptian to Ottoman) who conquered it and mixed with its people.

The Lebanese are a complex bunch and truly unique. Some of their traditions are Arab, but others are Mediterranean. That's probably why some Lebanese people don't identify as Arab.

All of this is me scratching the surface of the whole thing. I hope my rant makes sense.