This is partially accurate. To make sure you don't run into this issue again, I'd like to fully explain it.
The contents of your SD card, ALL your saves, are encrypted to your particular 3DS by a file stored in NAND called movable.sed. This is what lets you back your SD card up and format your card and then put the files back on and still have them be readable, but if you try putting your SD card into another 3DS they aren't, because the encryption on the card does not match that 3DS's movable.sed.
On top of this, movable.sed is tried to your NAND, not your 3DS, meaning a system format changes your movable.sed. So if you remove your SD card and format your 3DS, then put the SD card back in, it won't be able to read it. This is why save managers like JK's Save Manager can only export saves on the 3DS they were created on. Once exported they can be imported on any 3DS though.
One final thing, some games have a secure value, like Pokémon and Animal Crossing that adds an additional security that prevents you from using an older backup you copied even if you're using it on the same 3DS. JK's Save Manager will adjust this secure value as well.