How do I identify the model number of this old Mac?

That's a 17" CRT Graphite Studio Display and a Power Mac G4 tower, which can be one of 4 models produced between August 1999 and July 2001. On the back of the tower, you'll see the label just below the power supply connections indicating the model #, something like M7893. If you look at the arrangement of the ports on the back and compare them to this photo, you'll be able to identify the correct series: (1) PCI Graphics, (2) AGP Graphics or Gigabit Ethernet, and (3) Digital Audio. The specs for each model are listed here.

Thanks to Ron JACKLE...

Physical Serial Number...

If the iMac is a Tray Loading the SN is located in the I/O compartment on the frontmost flat boss near where the audio ports are.

Slot Loaders have their SN located on the belly along with a memory door.

...Ron

iMac G3...no serial number - Apple Community

Thanks to A Brody...

iMac G3:

http://docs.info.apple.com/article.html?artnum=58669

An additional important identifying factor for iMac G3s is whether or not they have built-in Firewire ports. This will among other things tell whether Mac OS X 10.4 can be installed on it, whether the Migration/Setup Assistant can be used with it, and whether you can make a bootable Mac OS X backup of it. This article helps tell whether or not they have Firewire ports:

http://docs.info.apple.com/article.html?artnum=58207 . A good way to tell if your iMac has Firewire, the ports on  describe exactly how the ports appear in your machine. If you only have USB ports, and no Firewire ports there, then you don't have an iMac DV.

https://macmaps.com/macosxnative.html#IDENTIFY

I doubt Apple still posts Serial# info for what looks like the 1st iMac G3, you may need to look it up here...

https://everymac.com/ultimate-mac-lookup/

https://everymac.com/systems/apple/imac/index-imac.html