BUM is understandable becaues all "Mage" jobs, Red Mage, White Mage, and Black Mage, have M as the last letter of their abbreviations (they aren't really acronyms), thus BUM following the same principle. BUT, if you notice that all job abbreviations use now vowels, only consonants to make the abbreviation "sound" more like the full job name, which makes BUM and BLU both questionable. However, since BL are shared by both Blue Mage and Black Mage, the only option SE would have if they were to follow the same train of thought would be BMG. This would make less sense because "mage" would be more represented in general than "blue", so they are then left with BUM, BLU, BUE, BEM.
Which one sounds the closest so Blue Mage? That's right: BLU.