That would normally be called a "flash controller" and yeah, of course that lives on the SSD.
(unless it doesn't - eMMC doesn't normally have a flash controller and you just do direct writes to the flash cells... as do some IOFusion PCIe cards that are just flash directly attached to PCIe with no controller. Sometimes that functionality is just software-based. Flash cards (eg microSD) usually also do not have a flash controller directly either.)
Anyway, it's true though that Apple does push the flash controller functionality into the SOC while AMD does not, Apple implements their SSD in a non-standard fashion, that's why it's not compatible with off-the-shelf drives. The flash is just flash on a board, I don't even think it's NVMe compatible at all either.
So if you want to be maximally pedantic... neither does Apple implement onboard NVMe controllers, just flash controllers ;)
FYI, all current flash card formats have the equivalent of an SSD controller, implementing a Flash Translation Layer. Exposing raw NAND was somewhat viable in the very early days when everything was using large SLC memory cells (see SmartMedia and xD-Picture Card), but no current format could remain usable for long without wear leveling. If you can use standard filesystems and access the drive or card from multiple operating systems, then it must be providing its own FTL rather than relying on the host software for wear leveling.
The above also applies to eMMC and UFS storage as found in devices like smartphones.
(unless it doesn't - eMMC doesn't normally have a flash controller and you just do direct writes to the flash cells... as do some IOFusion PCIe cards that are just flash directly attached to PCIe with no controller. Sometimes that functionality is just software-based. Flash cards (eg microSD) usually also do not have a flash controller directly either.)
Anyway, it's true though that Apple does push the flash controller functionality into the SOC while AMD does not, Apple implements their SSD in a non-standard fashion, that's why it's not compatible with off-the-shelf drives. The flash is just flash on a board, I don't even think it's NVMe compatible at all either.
So if you want to be maximally pedantic... neither does Apple implement onboard NVMe controllers, just flash controllers ;)