Sully ranks #842 with 140 male registrations. The name is short for Sullivan (an Irish surname meaning roughly "dark-eyed") and on a pet license usually carries the same sound-friendly register as Murphy or Finn: Irish heritage worn lightly.
The Irish-surname diminutive cluster
Sully sits with Murphy, Finn, Brody, and Duffy in the cluster of Irish-surname diminutives that work better as pet names than as human names. The source surname Sullivan carries strong Irish-American identity, and the diminutive form lands warmly on the family dog without the formality. Monsters Inc. (the Pixar film's main character James P. Sullivan, called Sulley by friends) added a younger-millennial layer that lands particularly on big fluffy dogs: bernedoodles, golden doodles, and Old English sheepdogs whose blue-purple-fluffy register matches the cartoon character.
Sound and breed lean
Two syllables, front-stressed (SUL-lee), with a soft sibilant opening and a gentle -y close. The name calls warmly outdoors and tolerates the babbling household register. Sully also pulls a smaller cluster from Captain Sully Sullenberger (the pilot of the 2009 Hudson River landing) which lends a calm-hero layer for some owners. See sheepdog names for the cluster fit.
The counter-reading
The honest concern is that Sully reads slightly more human than its Irish-cousin Finn or Murphy. The human Sully page shows climbing modest SSA presence. Households who want the Irish register with stronger pet-only identity might consider Finn or Murphy.