The Dog Agency describes itself as "home to the most influential animals in the world", but it's probably better known as the first talent management agency for animals. The Dog Agency aids their famous animal clients in developing their brand and creating content, and have worked with major companies such as Refinery29, Purina, Buzzfeed, Google, and more.
Pet owners can profit off of their pet's Instagram accounts through corporate sponsorships, although the money they make varies depending on follower count, engagement, and the brand that they're advertising. According to Mashable's article, "Inside the Glorious and Lucrative World of Instagram's Famous Pets", "...generally those with a couple hundred thousand followers are making $3,000 to $5,000 per piece of sponsored content, while those in the millions make $10,000 to $15,000."
Some pets who have large enough followings can also sell their merchandise to fans: