Well, centuries ago, a female donkey was called a jenny-ass and a young female owl was called a jenny-howlet. In old nursery-rhymes, Jenny Wren was thought of as the wife of Robin Redbreast, which makes the human-name connection pretty clear. To be honest, I can't see any connection between donkeys and trees or any species-specific connection between trees and wrens rather than any other bird.
The human name, Jennifer, is a Cornish version of Guinevere meaning 'fair' or 'soft' in old Welsh.