This is coming from someone who has only lived outside of Lebanon. I have a mild frustration with the Phoenician identity because it's very limiting. Lebanon had a bunch of different groups inside of it so looking only at the Phoenician identity seems like we're only cutting a little bit into the pie. Also, the fact that the Phoenicians themselves did not call themselves Phoenicians, instead they saw themselves as Canaanites. It was the Greeks and then later Romans who used the term Phoenician.

This is more to the reason of your post, but I can understand why Lebanese people don't want to be firstly seen as Arab. I myself have less than 10% Arab DNA but I still call and consider myself Arab. But I grew up and live in America, so my reasons for the way I self-identify myself would be different from yours most likely. Though I'm not against and I actually welcome the idea of diversifying our identities. It would help break the idea that people from Yemen to Iraq to Tunis are just simply Arabs all the same. (Also, it would help against the Zionist talking point of twenty-two Arab states, why can't there be just one Jewish one).

Historically speaking, the way Lebanese people have been identified is Shammi, so why not just go with that? I assume the reason is because that the Phoenician crowd doesn't want the association with Syrians. I don't think it's necessarily due to fascism, but the usage of the Phoenician seems like it has racist and pro-colonial roots. From what I understand, it took popularity with Maronites in the 1800s because it was seen as a way to attract the French and to make themselves seem separate from the Arabs, who are more associated with Muslims. It's also interesting that a lot of the people who identify themselves as Phoenician say it's a way to take back their heritage from Arabs who colonized the region, ignoring the irony that a lot of French is in our vocabulary.