Can changing my birth control pill make my nipples sore a month later? Could it be stress?

Yes. I doubt stress will cause your nipples to be sore, although, stress can be a good masquerader, so if nothing else is found to blame, you may as well do it for stress. Changing birth control pills can definitely cause sore nipples as there's a hormonal change that nipples will be accommodating to and soreness can be a result of it.