Screenshots on imgur because I could not attach three screenshots to this post:
Reproduced in Skirmish with maximum resources and “inna jiffy” cheat for faster build / research times.
No special steps were taken outside of building an archery range, a House of Wisdom, training an archer, and researching Composite Bows.
Assuming the tooltip reflects the attack speed accurately, either feudal archers are too slow, or castle / imperial archers are too fast. But nothing indicates that they would be intended to have different attack speeds.