Is Family Share good for backing up two family members computers in Time Machine? Or should each person have their own icloud account?

I may want use time machine with icloud. My wife also has a computer. Can we share the Family Share account for backup? Or is that for sharing photos and files between computers? Do we each need a separate icloud account to use time machine backup?

I have been using a WD external hooked up to a Airport Extreme for Time Machin backups but I just got Google WiFi. There is no way to plug the WD external into the Google WiFi. I tried a USB to ethernet connector with an ethrnet switch and it's not working. Going to the cloud might just be easier.