Cis people run into this problem too. My dad, his brothers, and my brothers all can pass for much younger than they really are. People think my dad is our brother when we're all out together. When he tells them his real age, they're shocked. Granted, he could never pass for 13, but people tend to think he's in his mid-twenties when he's really almost 50. This is not an exclusively trans male problem.
T helps though. People don't act shocked when I tell them I'm 19 anymore, although I personally feel I still look at most 15. Honestly, it could have been worse for me. There's a much smaller difference between 19 and 13 versus 35 and 13. It did feel deceptive when I passed for a younger guy but 13 wasn't that long ago for me, so ultimately it's not a big deal. I can't imagine how tough it is for people who transition later in life.
As far as automatically getting the kid's discounts on movies and the like, I really don't see a problem with that. You deal with the ticket person for all of 30 seconds, and you get to pay cheaper rates, so why not enjoy that while it lasts?