The claim here is that the historical Israelite kingdom of David, which lasted around a hundred years ~1000 BC and is the historical focus of a lot of stories in the Hebrew Bible, "controlled" various parts of what is now Southern Lebanon, either formally (very unlikely) or informally through hegemonic trade domination. The latter is potentially true, but how that then "proves" that Southern Lebanon is Israel is rather suspect, given that the historical Kingdom of Israel again lasted only around 100 years, was kind of a freak accident because of the giant power vacuum that developed at the end of the Late Bronze Age into the Early Iron Age in the Levant and the fact that Assyria had basically destroyed all of Jerusalem's competitors for them. The kingdom was later destroyed by the Neo-Assyrian Empire and later the Neo-Babylonians.

The claim is very funny because really, if you want to go like back further, "Israel" and "Lebanon" were both possessions of New Kingdom Egypt around ~1300 BC, and under Hittite domination before that, and before that a kind of merchant zone that shuffled between Egyptian and Sumerian control. So really, by Israeli archeological standards, Israel is actually Egyptian and Egypt has the historical right to conquer and exert control over their historical possessions in the Levant.