Cesar ranks #844 with 139 male registrations. The name is the Spanish form of Caesar, and on a pet license it lands almost entirely from one of two sources: Spanish-speaking households using the heritage form, or English-speaking households inspired by Cesar Millan, the celebrity dog trainer.
The Cesar Millan effect
Cesar Millan's National Geographic show The Dog Whisperer (2004-2012) made Cesar one of the most recognizable English-language pet-naming references of the 2000s. The name on a US pet license correlates strongly with households that watched the show and absorbed its training register. The naming logic is rare among pet names: an owner naming a dog after a person who specifically trained dogs. See pit bull names for the cluster Cesar Millan most often worked with.
The Spanish-heritage register
Separately, Cesar is a top-tier Spanish-speaking masculine name and lands on Hispanic-American household pet licenses with the same naming logic that drives Cesario or Diego: family-language continuity. The Spanish pronunciation (SEH-sar) differs from the English (SEE-zer), and call-name use varies by household.
Sound and the counter-reading
Two syllables, front-stressed, with a soft sibilant opening and a rolled R tail (in Spanish) or a flat R (in English). The name calls clearly outdoors. The honest concern: Cesar carries strong human-coding (the human Cesar page shows substantial SSA presence in Hispanic communities), and the dog will share call-name space with substantial human numbers. Households who want the Latin-classical weight without the human overlap might look at Maximus.