Is it possible? Not in the usual way it is portrayed on the internet. In those that are not photo-shopped, they have done one of two things: they have removed part of the penis shaft that is inside, reconnected the plumbing and the blood supply and then shoved a vagina in there using intestinal tissue and skin graft. They evidently can still get an erection but it is like a tree without roots and is quite bendable at the base. Then, the vagina has no nerve endings so there is minimal sensation. The other option is to turn an oversize clitoris into a penis. Yea, sensations everywhere but not the same as having both and it is difficult if not impossible to orgasm from penetration. So, this isn't practical if you want to actually enjoy sex as a male and as a female in the traditional lock and key kind of way.
However, there are a few, a very few, full hermaphrodites (certain chimeras, 46XX/46XY) who have most if not all of both sets of organs. (The penile shaft and prostate are moved forward making room for the vagina. Again, there is a problem. One woudl be running T which suppresses E to the point of no menstrual cycle and ultimately can cause serious problems with the female system. The other is running E, which suppresses the T. One wins, one looses.
Running both? Maybe, for a while, but it would be like switching from high doses of T for two weeks, then to high doses of E for two weeks, then to high doses of T for two weeks, then to high doses of E...it would be a two week cycle. That person would have serious cyclic mental issues – every two weeks for 24-36 hours. Then, the E, because of how high it gets at ovulation, over time, would actually start a deterioration process in the prostate and sexual function would go right out the window. Ultimately, the testicles would fall asleep and the male system would become worse than useless.