Derek at rank 1409 is a solidly mid-century American name doing full-formal service on a dog. It sits in the gap between Eddie and Edward, between Bobby and Robert. Derek never quite became a nickname the way many others did, so the pet-naming deployment doesn't have the usual formal-vs-casual tension. Derek is just Derek.
The Full-Name-for-Pets Pattern
Derek is a Dutch and Low German form of Theodoric, meaning ruler of the people, a name with considerable Germanic history that has been thoroughly domesticated in American English into something comfortable and unpretentious. On a dog, it doesn't read as formal or comedic the way Nicholas or Leonard do. It reads as simply the dog's name. Labs and Goldens are natural carriers.
The Derek Zoolander Angle
Derek Zoolander, Ben Stiller's magnificent protagonist in Zoolander (2001), is the name's most culturally resonant current association for owners under 40. That comedic layer is gentle enough to enhance rather than undermine the name on a dog. The human name's history is at /names/derek. Terry and Gary occupy similar comfortable territory.
The Counter-Reading
Derek's comfort is also its limitation: it doesn't generate a reaction. For owners who want the same masculine directness with more presence, Duke or Ranger push harder.