No, they do not all use the same type of battery. The Warrior tier uses a removable 18650 Li-Ion cell. The power rating on them has to be fairly high, the store page has the power specifics:
https://saberforge.com/collections/diy-electronics/products/18650-battery-1
Aside from those, there are also some nice TrustFire ones out there with the correct power requirements. As for chargers for those, SF sells a cheap one or you can source better ones pretty much anywhere that sells 18650 Li-Ion cells.
Both Veteran and the Hero setups will all use an 18650 with the same power requirements as the removable cell, but it’s permanently wired in and can’t be removed without soldering. Like the Warrior tier batteries, these cells also have a protection circuit built in.
The Champion tier (SaberCore 3.0) also uses a non removable 18650 Li-Ion cell. However, the power rating is higher and it has a fast drain and no protection circuit. These features are necessary for the board to function properly. The specifics for this battery are on this page:
https://saberforge.com/collections/diy-electronics/products/champboard
Getting the right battery for your setup is critical, though the sabers will come with batteries, except Warrior Mystery Boxes unless you pay extra for a battery and charger. The best charger to get for the Veteran and Hero tiers are the ones SF sells:
https://saberforge.com/collections/accessories/products/smart-charger-2-1mm-3a
As far as I know, the Champion SC3.0 just uses a standard USB-C to full size USB cable and charges via your computer’s USB ports.