The serial number on the Bambu Lab machine will be printed on a small sticker with the serial number and QR-Code which can be scanned by many smartphones when in photo mode. The serial number can also usually be found in the about section of the printer's menu as well as the outside of the box packaging.
For more Serial Number info from Bambu Labs: https://wiki.bambulab.com/en/general/find-sn
Bambu Labs P1 Series and X1 Series have the serial number sticker situated inside on the upper right side corner.
Bambu Labs A1 Series have the serial number sticker situated visibly on the rear exterior of the machine.
Bambu Labs H2D has the serial number situated inside on the right panel corner.
The Original AMS has the serial number sticker situated inside on one of the dividing walls.
Some deviations from these locations may occur.
Images provided by Bambu Labs