I'm not quite sure where the origin of the use of this word 'facteur' came from.
Latin roots as factor and it also has the same meaning and sense as with the English word.
But as a name for a postman I don't know.
After all there is a 'burueau de poste' so why not 'homme (ou femme) de poste'
Children here play many games (even on horseback) and sing songs associated with 'monsieur le facteur' ou 'Mme la factrice'. An important part of French culture just as the postie in Great Britain.
Be careful with female version I once called our post lady 'factice' missing out the 'r' thus meaning 'dummy'.