I have an advanced degree in musculoskeletal tissue engineering and have been working in the field for a while now. While I'm no doctor I might be able to help you a bit.
First 15 is pretty young to "stop growing" everyone is different but during adolescence we have something called an epiphyseal plate which is basically a line of cartilage cells that replicate elongating your bones and eventually turning into mature bone. The epiphyseal plate is more commonly known as a "growth plate" in layman's terms. Once someone is finished with their growth cycle this plate fuses or simply goes from being cartilage into being bone. That's when someone says you're growth plate has "fused."
What you need to know right now is if this has happened. Adolescents treated with HGH will grow longer bones than those who don't but this is only possible if your "growth plate" has not yet fused.
My recommendation is to ask for an X-ray to ensure that his plate is fully fused and therefore he truly has "stopped growing." Also it might be a good idea to get a second opinion. If a doctor doesn't see an epiphyseal line in an x-ray than there is still potential growth. That growth can be potentiated by treatment with HGH.
I am 5'6" and so completely understand what your brother is going through. He's probably fairly tall for his age right now. I grew really fast in middle school and throughout high school most of my peers passed me up. It's one reason why I work out so much. A V shaped body makes you appear taller than a pear shaped one.
in any case, request an X-ray and get 2 opinions. If his plates are fused then he's pretty much done growing. If not he can ask to be treated with HGH. The difficult part is that 5'6" is still within a normal height range and most doctors won't prescribe treatment unless he's in the lowest 10th percentile. Meaning he's got to be less than 5'2".
Hope this helps.