After a lot more back and forth with Bosch support, I finally got an actual answer.
The serial number on a Dremel is written on what they refer to as the "data plate" on the side of the tool. The number doesn't match what's in any of the QR codes on the tool, and it isn't labelled as being a serial number.
It's a 3-4 digit number, followed by a date of the form MM/YYYY
.
If the first number was 179
and the date was 02/2024
, then the serial
number of the tool would be, apparently, 0179022024
.