The first animal to die on the Earth after sin entered was killed by whom? "Also for Adam and his wife the Lord God made tunics of skin, and clothed them. [Genesis 3:21].
On the Earth, the first chapter of the struggle between Good and evil was written in the blood of an animal; probably more than one, given that there were two standing naked in their sin. If Biblical typology can be retrospectively applied in this situation, I will posit that the animal(s) that were sacrificed to cover the nakedness of Adam and Eve were...lambs.
Given the symbolism of the lamb...the Lamb of God which takes away the sin of the world...the Good Shepherd...the sacrificial service the Lord gave Moses...and the followers of Christ as His flock: we see the lamb as a recurring symbol of Christ; and it began in the Garden of Eden.
The death of the first animals occurred in a world newly fallen into sin. Killing the animals prefigured Christ as the Lamb that would die for the sins of the world, and their skins (tunics) represent the robe of Christ's righteousness that we wear because of His death in our place.
Adam and Eve did not die (immediately) for their sins; it was innocent lambs that died so that they could be covered. The same is true of our sins; our Innocent Savior died to pay the price for our sins. Before the beginning of the life on this Earth, He had us covered! ec