Apple doesn't seem to be ruling out Ventura AE Time Machine usage yet (or just hasn't updated this to say so yet):
https://support.apple.com/en-us/HT202784
And of course, you should open Airport Utility on both Macs to check for software updates for AE. There are occasionally such updates so maybe you are not up to date on yours. Open on both High Sierra and then Ventura to see if either wants to update the AE.
I bailed on AE a few years ago but I THINK I remember a time when I attached a USB hub to the USB jack and it worked. If you have a hub laying around, you might try it with the High Sierra setup that already works. If it still works through the hub, try the option lower on the page at the above link: direct connect a
second external drive and select it as a Time Machine backup. Backup. Eject and then go attach to the USB hub. See if you can then select it as a TM drive through the AE.
If that doesn't work, format the second external in the older file format, hook it to the Ventura Mac, select as Time Machine backup and see if you can get it to backup with reformatting the drive. If so, backup, eject, take it to the AE and hook it to the hub. Then see if you can select it as TM through AE.
Net result would be having 2 externals hooked to the AE: one for each Mac.
I really can't recall this working but it's easy enough to try if you have the pieces laying around.
When I gave up on AE, I switched
TM needs to the Synology network drive option and it works very well. My new router has a USB jack but I haven't even tried anything attached to it. Synology works fine and with both old and new Macs. Consider that option if you would like to replace the aging router.