Most people in the sixth century did not have surnames. Instead, they were often identified by a nickname or were named after their place of birth/base of operation. We have for instance bishops called Timothy the Cat, John of Ephesus, and my favourite, Athanasius the Camel-Driver.

As far as I can tell, no sources explain why Paul was called 'the Jew', though Pauline Allen has suggested the he's called this because he was a persecutor of the miaphysite church, so in miaphysite Christians' eyes, he was akin to a Jew. Basically, after the Council of Chalcedon in 451 there were two main factions within Christianity. One held that Jesus had two natures, one human and one divine, but the two were united in a divine hypostasis (the Chalcedonian view), whilst the other held that Jesus only had one nature, aka he was fully divine (the miaphysite or non-Chalcedonian view). Paul was a Chalcedonian patriarch imposed on Antioch after the previous miaphysite patriarch was exiled, and he apparently alienated a lot of people through his harsh policies. He would eventually resign, perhaps because the opposition was so vociferous.

I think I would have to agree with that interpretation, since the two miaphysite sources that mentioned him, pseudo-Zachariah Rhetor and John of Ephesus, both called him 'the Jew' (and accused him of another heresy, Nestorianism), whilst Evagrius, a Chalcedonian ecclesiastical historian, did not. I can't access the other sources, which include a few papal letters and the Chronicles of John Malalas and Theophanes Confessor, as unlike the previous three sources, these Chalcedonian sources are not on my laptop as handy PDF files, so I can't confirm this theory for sure, but I think it is a reasonable enough suggestion.