No animal has served as an emblem of state as continuously as the eagle, and the line of descent is traceable rather than accidental.
The aquila was the standard of the Roman legion, a silver or gold eagle on a pole carried by the aquilifer. The reform that made the eagle the sole legionary standard is attributed to Marius in 104 BCE. Loss of the aquila was the gravest disgrace a legion could suffer, and the recovery of the standards lost at Carrhae was presented by Augustus as a major achievement of his reign. The bird was the bird of Jupiter, so the standard carried a religious as well as a military charge.
From Rome the eagle passes forward directly. The Byzantine and then Holy Roman use of the double-headed eagle, looking east and west, was inherited by Russia after Moscow claimed the Byzantine succession, and by Austria. The single-headed eagle was taken by Napoleon in conscious imitation of Rome, by Prussia and then Germany, by Mexico from an entirely separate Mexica foundation story, and by the United States in the Great Seal of 1782, where the bald eagle holds arrows and an olive branch. Benjamin Franklin's letter disparaging the bald eagle as a bird of bad moral character is genuine, but it is a private joke in a letter to his daughter, not a proposal for the turkey, and the story is habitually mistold.
The eagle feather in many Indigenous North American nations is a different matter entirely, a sacred object with specific protocols governing who may hold one and how, protected in United States law by exemption from the Bald and Golden Eagle Protection Act for enrolled members of federally recognised tribes. It is not an emblem of state and should not be discussed as one.