While Lebanese and Arabic are two different languages, the Lebanese (in Lebanon) do not worry, or may not care, for the difference between the two languages. They both come from Semitic roots, and have many common words and grammar.
The Lebanese more than likely wanted to distinguish the language they speak from French (a Latin language) that is widely used in schools and normal life, from English (a Germanic Language) that is growing in education and business life, and from Armenian (an Indo-European Language) that is used by some Lebanese in education and business.
Dealing with all these languages descending from different linguistic branches probably made it easier for people in Lebanon to refer to the language by the roots or by the script (the letters) they use to write it with. Since Lebanese in general use Arabic script to write Lebanese, they often refer to the Lebanese language as Arabic language.
Add to that that the Turks who occupied the region for four centuries used to call all the Semitic languages other than Hebrew – “Arabic”, since they did not differentiate between the spoken Lebanese, Egyptian or Arabic then.Some people tried to reach a compromise by claiming that the Lebanese people speak Lebanese but write Arabic. This incorrect statement often used by people in Lebanon who confuse the use of Arabic script to write, with the language itself.
The Lebanese use Arabic script to write both Lebanese and Arabic; They write, read and speak Lebanese for their songs, poems, television production and letters, while they write, read, and speak Arabic in Arabic Literature, courts and some formal religious rituals. This is not different from the people utilizing Latin letters to write both English and Spanish languages.
Since the languages that the Lebanese (in Lebanon) are familiar with are from different roots except for Arabic and Lebanese, they think that there are no two languages closer to each other than Arabic to Lebanese. Also, since the Lebanese are raised learning Lebanese and Arabic side by side, while the other languages they learn are from different roots they missed the opportunity to compare apples-to-apples such in comparing Italian to Spanish for example.
Linguists here, especially those who were not raised learning these two languages, can point out the difference. Professor Wheeler Thackston of Harvard pushes this analogy a bit further and argues that “the languages the ‘Arabs’ grow up speaking at home, are as different from each other and from Arabic itself, as Latin is different from English.”