Are the black and brown discharges that appeared forty days after childbirth, while taking birth control pills, considered menstruation, with all the rulings that entails?

The blood seen by the questioner after the forty-day period, exceeding 15 days, is istihada (non-menstrual bleeding), and she should have done what a woman experiencing istihada does. As for the recurring blood, it is not considered hayd (menstrual blood) by the majority of scholars unless its total duration reaches one day and one night and it occurs within a timeframe suitable for menstruation. If the questioner abandoned prayer during the days of istihada, thinking it was hayd, then she must make up the missed prayers.