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There is an Arabic male forename, Assi, and also a surname, presumably a patronymic. You'll also find surnames al-Assi (with the definite article), and Abu Assi ("father of Assi"). These names are found in various Middle Eastern, Arabic speaking nations, but I haven't found Assi as a surname in Lebanon, only as a forename.
As for Assi(s) as a French surname, I've found no evidence to date.
There is a close relationship between France and Lebanon, and some Lebanese Christians give their children French names, particlarly the Maronites and Catholics. They also romanise surnames along French lines (e.g., Qasis written as Cassis, Jurjus written as Georges), which might result in a Lebanese name being mistaken for a French name.