He didn't just want to go hunting, he saw himself consistently as a boy. A lot of girls want to do "boy" things, but their arguments tend not to include the phrase "I'm a boy"
My understanding of Japan is that they're honestly pretty conservative (you notice Chiaki was sterilized during his gender confirmation surgery, that's basically mandatory in Japan if you transition) on issues and so an old doctor using outdated terms isn't surprising.
This is a really well done story. I'm glad I stuck with it after the first couple pages (when they said Chiaki was a man, I was concerned it was a crossdressing story, which isn't my thing)