The confusion appears to stem from identifying Thoros I, son of Constantine with Thoros of Edessa, the latter of whom Gabriel is attested as being the father-in-law. Gabriel must have had some connection to the Greek culture, either via his mother or wife and, if that connection was to the family of Constantine I, it was most likely further back.
The confusion stems from associating Thoros I, son of Constantine, with Thoros of Edessa, who is attested as Gabriel's father-in-law. It seems Gabriel had a cultural link to Greece, possibly through his mother or wife, and if that connection was to the family of Constantine I, it would probably be more remote.