According to the OED,
to be born and bred, or bred and born: an alliterative phrase in which bred has usually sense 9, though formerly sense 1.
1542 N. Udall tr. Erasmus Apophthegmes f. 133v
In the same Isle born, breden, and brought vp.
a1616 Shakespeare Twelfth Night (1623) i. ii. 20
I was bred and borne Not three houres trauaile from this very place.
Sense 9 is that of animal husbandry, whereas sense 1 is of
trans. Said of a female parent: To cherish (brood) in the womb or egg; to bring (offspring) forward from the germ to the birth; to hatch (young birds) from the egg; to produce (offspring, children).
This suggests that bred and born would indeed be more congruent with the chronology of gravidity and parturition, but binomial word order does not depend on such things, hence thunder and lightning. Or perhaps, in the reverse process of kith and kin, people came to associate the saying with a different sense of breed. Right after its entry for born and bred, the OED attests to breed meaning
- To train up to a state of physical or mental development. … b. To train up (young persons) in the arts of life; to educate, tutor, bring up. … †(a) To train by education, educate, teach. Obs. (b) To bring up from childhood, including all the circumstances which go to form the religious persuasion, manners, position in life, and trade.