Can my dog be more genetically diverse than I am?

If your dog is a mutt, almost certainly yes. If your dog is a purebred, almost certainly no.

Modern dog breeds were created mostly in the 19th and 20th centuries by selecting for a small set of physical traits and aggressively inbreeding to fix them. Many breeds today trace back to fewer than a dozen founder dogs. The genetic diversity within a single breed - say, Cavalier King Charles Spaniels, or German Shepherds - is brutally low. Some breeds carry inbreeding coefficients equivalent to first-cousin marriages, in every individual.

A street dog or village mongrel, by contrast, has been mixing freely for generations and is wildly more genetically diverse than its purebred neighbours.

Humans sit in between. Within a single ethnic group, genetic diversity is moderate. But humans as a species are unusually low-diversity for a mammal of our population size - we went through a population bottleneck around 70,000 years ago and the species has expanded from a small founding group since then. There is more genetic variation within a single troop of chimpanzees than within the entire global human population.

So: your indie mutt is probably more genetically diverse than you are. Your friend's purebred Pug almost certainly is not.