The bireme is perfectly appropriate as a UU for a Phoenician civ. The Phoenicians invented it (and the trireme). They were prolific shipbuilders and sold many to others across the Mediterranean, which is partly why the bireme became ubiquitous.
Bireme as a Phoenician (city-states) invention is pretty solidly established, but the Trireme not so much: there is still on-going debate whether it was a (rather natural) development of the bireme by the Phoenicians or the Greeks, since it spread so fast that it largely replaced both monoremes and biremes as the standard warship all over the eastern Mediterranean and as far west as Carthage.
The fairly well-established origins of 'iconic' classical ships are:
Pentekonter ("50-oared") - the ultimate monoreme, mentioned by Homer in the Iliad and thus attributed to the Greeks
Bireme - archeological and written evidence attributes it to the Phoenician city-states of the Levant
Trireme - Either Phoenician or Greek, but whichever it was the other adopted it virtually instantly.
Quadrireme - invented around 400 BCE in the Ortygia workshop in Syracuse, the first of the larger-than-trireme Polyremes
Quinquereme - origins somewhat cloudy, but appeared within 50 years after the Quadrireme in Carthage and Greece and quickly replaced the Quads and other smaller ships because the Quinquereme was a strong and stable enough hull to mount small catapult bolt-throwers - the first real ship-borne 'artillery'
Deceres (or Decireme) - the largest Polyreme known to have been used in actual battle: at the culminating naval battle of Actium, the flagships of Anthony's, Cleopatra's, and Octavian's fleets were all Deciremes.
Liburnian - the 'classic' combination type: used both oars (bireme configuration) and sails, was invented by the Liburnii of the southern Dalmatian coast and adopted in various forms by pirates, Imperial Romans, and smugglers