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.