Great question — answering as the UphillTech team:
No, there's no additional documentation or security you pass through for Japanese domestic flights. In my experience, the only issue as a foreign tourist is that you'll have to pay more attention to what's happening at the gate as those announcements will likely be mostly in Japanese.
Full answer by pboss3010: https://travel.stackexchange.com/a/204640 (CC BY-SA 4.0).
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