The ordering of the letters is ultimately Phoenecian: the order of the Phoenecian alphabet was maintained in Greek (with a little bit of jumbling, and some leftovers appended at the end), and thence in Latin (again, with a little bit of jumbling—C G Z).
Why is the alphabet song easy to remember in English? That’s actually the wrong way around. Latin, and its successors, dropped the letter names of Greek, and went with letter names that (mostly) rhymed: a be ce de e ef etc (Latin alphabet).
Once the letter names rhymed in Latin, an easy to memorise alphabet song was inevitable in the languages that inherited its script and letter names. We certainly don’t have one in Greek.