Ultimately it's a sheer coincidence that the two have the same name but a very interesting one at that, Tripoli (Libya) was first founded as the colony of Oea by the Phoenicians, who came from what is now within modern Lebanon.

Tripoli (Lebanon) was also a Phoenician settlement as well under the name Athar.

Both cities eventually came under Greek and then Greco-Roman influence and it is through this that the modern variant Tripoli from the Greek Tripolis ( three cities) comes from.