Under arm bulge - is it a fat tissue or axillary breast?

Your description is completely consistent with axillary breast tissue. Physical examination would be necessary to substantiate this. If this ends up after the breasts are physiologically stable to be in surplus it can be reduced by liposuction or direct excision.

What you describe bests fits breast tissue that enlarged during pregnancy and breast feeding. You may have some mastitis from breast feeding which should settle down. In some women, the excess tissue goes away. In other women, they keep the excess tissue. Just fatty tissue can be removed with liposuction. Redundant breast tissue and skin may need to be excised. Give yourself more time to let your tissue settle. If after a few more months it has not gone away, I would suggest meeting with a board certified plastic surgeon to discuss your options.