Origins
A delivery driver and a midwife share the same etymological ancestor. The word deliver comes from Old French delivrer, from Latin līberāre — 'to set free'. Its oldest meaning in English is liberation.
The Latin līber meant 'free', and from it grew one of the richest word families in English: liberty, liberal, liberate, and deliver. The connection between delivering a package and delivering a prisoner is the transfer of something from one state to another — from held to released, from sender to receiver.
The childbirth meaning is the most ancient. To deliver a baby is to free it from the womb. Medieval midwives spoke of delivery in exactly this sense, and the term survived unchanged into modern medicine.
Later History
The postal meaning — handing over goods — developed from legal language. In medieval law, to deliver property was to transfer it formally from one person to another. The postman's delivery is a legal act disguised as logistics.
Even deliberate belongs to this family. Latin dēlīberāre meant 'to weigh freely' — to consider something without constraint. To deliberate is to think as a free person.