If the reason we die is because death is the wages of sin, why do animals die? Did they sin also?
The connection between sin and death, which you cite from Romans 6:23, is just one explanation for death that we might consider. If we look across history, many explanations for death have been offered. From a modern scientific point of view, we would point to the natural, biological limitations of organisms, from humans to animals to plants, as causes for death, not sin. Put differently, living organisms were always going to die; death is a natural part of life. In specific instances of death, of course, we would also cite many human-related factors as well as natural disasters, accidents, and other misfortune.
Within the Bible, there are a number of explanations for death, some different from that in Romans. Some texts, like Psalm 82:7, take death as a defining characteristic of humans—in distinction from deities, who are understood to enjoy immortality. In this instance, there seems to be no direct connection between sin and human death. In other texts, humans and animals are understood to be equally accountable for their actions and thus subject to punishment, including death. Genesis 9:5–6 reflect this view. These verses insist that God will deem any living being—animal or human—culpable for killing a human. (Important to note here is that many English translations render Gen 9:6a incorrectly: it should read, “Whoever sheds the blood of a human, for the price of that human their blood shall be shed…” Confirming this reading is v. 5, where God says explicitly that he will be the one to hold killers, whether animal or human, accountable.) This Gen 9 rule accords with the rationale for the Flood described in Gen 6:11–13, where “all flesh”—including animals—are said to be engaged in the violence that the deity finds intolerable and that prompts the sending of the Flood. Other biblical texts seem to treat animals as belonging to an entirely different category from humans and thus not culpable for their actions. In these cases, their deaths are apparently taken to be part of the natural order.