When biologists talk about evolution, they say the phrase "common ancestor" a lot. This is usually an unidentified organism that eventually gave rise to both (or all) of whichever modern species that the biologist is talking about.
Usually, I don't think people internalize what this means. It's not just that an organism with some common traits subdivided until its descendants were unrecognizable to each other; it means, quite literally, that they have the same ancestor.
The tree of life is a giant family tree.
You and your dog share a common ancestor. We don't know exactly what animal it was, although we can make some guesses about what it was like. It was a vertabrate, with a spine and jaws and bones. It was a tetrapod, so it had four legs. It was warmblooded. It was a placental mammal, so it gave birth to live young, fed them milk it produced, and was covered in hair. It was also the common ancestor to most other placental mammals, because carnivores and primates aren't that closely related to each other as mammals go, so it was probably small and kind of rodent-y (but not actually a rodent).
We don't know precisely which species this ancestor was, but more than that, we also don't know which specific animal in that species that this ancestor was.
And it was a specific animal. At some point, there was a last time that your ancestor and your dog's ancestor were ever siblings, and those siblings were specific creatures who had a specific parent.
You have parents and grandparents and great grandparents, and your dog has parents and grandparents and great grandparents, and all of these grandparents have grandparents who have grandparents, back and back thousands and millions of years, until the grandparents aren't dogs or people but wolves and intelligent apes, then basal primates and carnivores, until the great-times-a-million grandparents share very little resemblence to you, but are nevertheless your great-times-a-million grandparents.
And eventually, if you go far enough back, there will be a small and rodent-y creature that gives birth to a second small and rodent-y creature, and that child is your dog's great1000000 grandparent. And that first small and rodent-y creature will give birth to another small and rodent-y creature, and that child is your great1000000 grandparent.
Maybe your great1000000 grandparent and your dog's great1000000 grandparent were littermates, and rolled around together against their mother while they were learning to see and walk. Maybe one of them was born in another season, and their only commonality was the fact that they shared a parent or two. Or perhaps this species' older young helped take care of their younger siblings (probably unlikely, but who knows!), and either your or your dog's great1000000 grandparent fed and played with and groomed the other one.
And then they both had children who had children who had children who eventually gave rise to every dog on planet Earth and every human on planet Earth, and now after millions of years, your family is reunited in your living room.
The exact same story is true about you, and the tree in your backyard. You would have to go back much further, and that ancestor certainly didn't have littermates, since it wasn't an animal, but nevertheless there is a last time that both of you shared a great1000000 grandparent, and its children eventually became every plant on Earth, and every animal on Earth, including you, and including the tree in your backyard.