I also am just beginning to think it's a sign of an amusing/depressing tendency originating with Klaus, of seeing family as far more totally controlling than even Tarvek with his aristocratic bias does. Between writing off all Mongfish (with the possible exception of Theo) and all Sturmvoraus...

Klaus is beginning to sound prejudiced in the most fundamental sense of the word: he treats family as though it's a useful and reliable label or brand name sparing him the need to deeply question how and why each individual functions. Family allows him to prejudge. Gil choosing to distance himself as much as he does with the surname or the full name, but not just "Hey, Tarv," seems possibly indicative of falling into the same mental trap.