Swap Mac Pro (2013) SSD to another Mac Pro (2013)

Best practice requires a Full and Recent Backup. You can not rely on the drive making the transition without needing to be re-formatted, so you may need that backup. If it is not needed, Merry Christmas, you get an unexpected present.

It would be prudent to upgrade each Mac Pro 2013 involved to at least 10.13.6 before attempting this transfer, to get the updated (but backward-compatible) firmware in place. Since there is a non-zero possibility of damage to the Mac or the SSD, do this even if you do not expect to keep 10.13.6 or later for your daily use.

The third-party replacement drives available for this Mac are NVMe drives. All require that your Mac ALREADY be running 10.13.6 or later. The firmware update required for 10.13.6 or later requires the original Apple SSD drive, and can not be installed if the original drive has been replaced.

Yes you can transfer the original Apple SSD to another computer of the same model. You may need to change the default Startup Disk in System Preferences though for slightly faster booting.