FWIW, if you want an easy way to record the serial number of all your Alexa devices, you can find them under Settings in the Alexa app and just do a screen shot for each device. You can also find the serial number for all your Amazon devices in your account on Amazon's website--look under Manage Your Content and Devices.
The wouldn't have helped the OP since he couldn't get his Echo to be recognized, but it provides an easy way for those of us who have Echos that are already functioning to record our serial numbers. However, since the information is readily available in at least two sites, I'm not sure having an additional record is really necessary so long as you are the registered user/owner of the device.