Max has a traditional Roman tria nomina which is a three part name. The tria nomina consists of the praenomen, the nomen and the cognomen.
The praenomen is similar to a first name which is given by parents for life. Faustus was a popular praenomen because it's Latin for lucky or fortunate. Maximus is lucky because he got to keep his tail. Most corgis have their tails docked when they are just a few days old.
The nomen is a family name passed on from generation to generation. Bubulcus is the Latin family name of of a cattle herdsman. Because corgis were breed to be herding dogs, this is the name he took.
The cognomen is closer to what we would call a nickname. Maximus is Latin for the greatest.