My question is, are these adapters made for the US and Japanese regions identical? It says 100-240 V AC, as if you could use said charger would be the included product anywhere in the world.
Exactly. Switching power supplies nowadays such as laptop chargers are truly universal - they work anywhere in the world and adjust to the line voltage automatically within a very wide range of tolerances.
I worry because I feel like my old laptop, which I'm using now in Japan, has gotten it's battery run down by a charge that's not strong enough.. that's my feeling anyway.. I have to leave it plugged in or the battery's dead within 20-40 min.
Lithium-Ion batteries degrade over time whether you use them or not.
A faulty or underpowered charger wouldn't charge your battery at all and it would have a hard time just powering your laptop.