There's more to it than that. NVMe is a higher layer technology than PCIe.
SATA drives connect to the host system over a SATA PHY link to a SATA HBA that itself is connected to the host via PCIe. The OS uses AHCI to talk to the HBA to pass ATA commands to the drive(s).
PCIe SSDs that don't use NVMe exist, and work by unifying the drive and the HBA. This removes the speed limitation of the SATA PHY, but doesn't change anything else. The OS can't even directly know that there's no SATA link behind the HBA; it can only observe the higher speeds and 1:1 mapping of HBAs to drives. Some PCIe SSDs have been implemented using a RAID HBA, so the speed limitation has been circumvented by having multiple SATA links internally, presented to the OS as a single drive.
NVMe standardizes a new protocol that operates over PCIe, where the HBA is permanently part of the drive, and there's a new command set to replace ATA. New drivers are needed, and NVMe removes many bottlenecks and limitations of the AHCI+ATA protocol stack.
SATA drives connect to the host system over a SATA PHY link to a SATA HBA that itself is connected to the host via PCIe. The OS uses AHCI to talk to the HBA to pass ATA commands to the drive(s).
PCIe SSDs that don't use NVMe exist, and work by unifying the drive and the HBA. This removes the speed limitation of the SATA PHY, but doesn't change anything else. The OS can't even directly know that there's no SATA link behind the HBA; it can only observe the higher speeds and 1:1 mapping of HBAs to drives. Some PCIe SSDs have been implemented using a RAID HBA, so the speed limitation has been circumvented by having multiple SATA links internally, presented to the OS as a single drive.
NVMe standardizes a new protocol that operates over PCIe, where the HBA is permanently part of the drive, and there's a new command set to replace ATA. New drivers are needed, and NVMe removes many bottlenecks and limitations of the AHCI+ATA protocol stack.