Amie is a word, the feminine form of the word for 'friend', ami being the male version in French, amies the plural.
The name
Amy comes from the French name
Aimee, past participle of the verb 'aimer' meaning 'to love'. It began as a nn signifying 'beloved' in the middle ages and was gradually adopted as a given name.
The 'aimer' verb is the clear root in common but the meanings are different. In French you might say 'J'aime mes amies' meaning 'I love my friends.' In modern times with the fondness for mucking-about the spellings of names you find Aimee/Amy rendered in all sorts of outlandish fashions - Amee, Aimie, Aimy, etc. including by pure coincidence or a basic grasp of high school French, the form
Amie. But this isn't the etymology of the name, simply a cousin with similar roots ;o)
Devon