I hate to break it to you but if he is already balding that severly, he is past the growing phase. He is not going to grow any taller. hair loss does not occur concurrently with height growth. The pathways and hromones that promote bone growth also promote hair growth. DHT is androgenic and actually supresses Testosterone (High DHT means less T is produced, by inhibiting DHT, T is allowed to rise). Testosterone is anabolic, it makes things grow(bone, muscle). DHT is only needed in the womb and at the onset of puberty. Taking finasteride will increase his T and may actually allow him to grow more.
I started taking it at 19 and actually grew another inch and a half after not having grown since 14. Finasteride does not stop growth so long as your body does not naturally have high estrogen(in males this is really only possible with large amounts of body fat), it helps it by increasing T, the most anabolic male hormone. My male part also continued to grow into my 20s. Finasteride gets a bad rap but really, what you are doing, is trading a shitty androgenic and highly carcinogenic hormone and suppressing it's production in exchange for a health promoting anabolic one. Finasteride also increases progesterone, which antagonizes estrogen. With finasteride, you get high T, high Progesterone, and low DHT and Estradiol (as long as body fat is not an issue). In my unprofessional and anecdotal experience, high DHT and estradiol go hand in hand (just look at the results you posted on forum) because high DHT means high 5 AR which also converts progesterone to allopregnanolone. With high amounts of 5AR your T and Progesterone get converted allowing estradiol to rise. DHT does not stop estradiol formation, it does not inhibit aromatase enzyme, it just atagonizes it's effects in tissues. This doctor probably is seeing the same thing. He wants to lower DHT and estradiol based on what you are saying he is prescribing.
People on here talking about he is still developing have no clue what they are talking about. He is devloping but not in the way that is considered normal for teenagers his age. I would def take finasteride if I were him. Possibly dutasteride. The only thing about dutasteride is it inhibits type 1 5AR which is found in the brain and needed for converting progesterone into allopregnanolone. This could potentially affect cognitive function. Finasteride does not have this problem.
Another thing to consider is upcoming treatments. RCH-01. It takes cells from the hairs on the back of his head, multiplies them, and re injects them into the rest of the scalp effectively giving the hair on the top of the scalp the same resistance to DHT as the hair that the sample was taken from. How thick is the hair on the back of his head? I see the sides look thin too.... As long as the back is thick, he can get this treatment hopefully in the near future. It may be available in Japan in the near future.
Really though, it should be his choice. Have you asked him what he wants to do?