Find a good photo of your dog, swipe up, and you should see an icon of your dog there. Just tap that, add a name, and you should see your dog in the people and pets album
I will keep checking back though! Eventually it will show up (just like I’m sure that one day my Memoji will actually show on my AirPods animation one day, as was promised by apple).
Go to search tab in Photos and there should be a “People & Pets” section. Scroll through that till you find your dog’s bubble. Tapping that should bring up some results and an “unnamed pet” box at the bottom. Tapping that will bring up a profile of your pet, tapping the “Add Name” header will let you name your pet.