They're just different things. The FDIC insurance is for if the bank itself goes insolvent and they literally don't have enough money to cover their depositors' balances anymore. There's no reason a safe deposit box would be affected.
Yes, they are different things. A safe deposit box wouldn't be affected by the banks insolvency.
A safe deposit box may be affected by other things and if those things happen they don't have to "make it right", if you go to the courts and make your case you may find that they are not at fault and you are not owed any compensation.
Probably varies from bank to bank, but in my experience you have to specifically buy separate insurance if you want the content of your deposit box insured. The big problem from the bank's point of view is that, unlike your bank account, the bank doesn't know what you have in the box and thus has no idea what to insure it for and no way to verify any claim.
More importantly the bank doesn't profit more from something more valuable being in your safety deposit box so it doesn't make economic sense for them to be the one insuring it.
