Trees die too, and so do mosquitoes. Sooner or later, every living thing dies. They are parts of a universe that was doomed to "frustration" when Adam and Eve sinned (Rom. 8:20).
Your question asks why death claims innocent things? The answer is at least implied in the creation account in the first three chapters of Genesis. The account suggests that the earth was made for Adam and Eve, but they spoiled it. At first, they were put in charge, but when they disobeyed God by eating forbidden fruit they lost control, and the earth itself and all its living things were cursed "because of [them]" (Gen. 3:17-19). The damage done by the fall of Adam and Eve touched everything, animate and inanimate. In time, Adam and Eve died, and so did Adam's dog, if he had a dog.
The apostle Paul may have been pondering these implications of the creation story when he wrote his epistle to the Romans. Romans 5:12-14 says that death entered the world through one man, and spread to everybody, including those who had not sinned in the same way as Adam. That would include babies and toddlers and other living creatures, including dogs. Echoing Genesis 3, Romans 8 explains that the earth itself and perhaps the cosmos was affected by Adam's sin. God cursed the whole earth because of Adam's sin. That is why dogs die.