"The Twelve Days of Christmas" was written in England as one of the "catechism songs" to help young Catholics learn parts of their faith during the time of King Henry VIII.
The "true love" mentioned in the song doesn't refer to an earthly suitor, it refers to God Himself. The "me" who receives the presents refers to every baptized person.
The partridge in a pear tree is Jesus Christ, the Son of God. In the song, Christ is symbolically presented as a mother partridge which feigns injury to decoy predators from her helpless nestlings, much in memory of the expression of Christ's sadness over the fate of Jerusalem.
The pear tree is the Cross itself. (II Kings 5:24)