First of all, the Carthaginian Empire's army was mainly mercenary. They fielded Iberian Infantry, African Infantry, Numidian Cavalry, and the now extinct North African War Elephant. Iberian infantry were light troops, with sword and shield. Because there were so many mercenaries, there was not a specific Carthaginian insignia.
Hannibal Barca has tons of information about the topic of the Carthaginian army insignias.
Concerning the Crescent Moon, it is actually a horizontal crescent above a circle, based on regiment, just like the Romans did with their battle standards. It is a symbol for the goddess Tanit, who is associated with fertility, war, and it is possible that infant sacrafice has been committed in her name.
The Golden sun of the Sacred Band was also probably a well known symbol to the other mediterranean.
The original Phonecians were not exactly a military power, and the Carthaginians were Phonecians, so it would be wrong to say that Carthage had its own specific symbols.
