SP Applies to Every Character and Weapon
Once you reach the Sector 7 slums in Final Fantasy 7 Remake, Biggs will offer to mod Cloud’s Buster Sword, which means that you’ll finally be able to start upgrading your weapons.
To upgrade weapons, you need SP, which is gained by leveling up in battles. From your main menu, you can then spend SP on weapon upgrades to unlock stat buffs and abilities for specific weapons. The important thing to remember here is that every weapon and character will get the same amount of SP and they’re all separate, which means that you don’t need to worry about saving your SP for one character or weapon, and you can upgrade everything at the same time.
In addition to that, it’s also possible to reset your weapon upgrades. In the Sector 7 slums, after you unlock the ability to take on side quests, you’ll meet an NPC named Chadley. Talk to him, and you can pay him some Gil to reset upgrades for a weapon, and you can respend your SP.