How do I backup my wife’s iPhone to our family shared iCloud account?

I'm not sure what you mean by that. Each person should have their own Apple ID and use that on each of their devices. iCloud storage (5GB free) comes with each Apple ID. If she is sharing your iCloud storage then she has access to the same amount of storage as you do.

If you are using Family Sharing and have elected to share iCloud storage, it doesn't matter whose account is the one that shared that storage. Each person continues to access the storage and to backup to iCloud using their own Apple ID in exactly the same way whether they are sharing storage or not sharing storage.

Her iPhone doesn't "find" any other account -- it uses her own Apple ID for iCloud in exactly the same way whether or not she is sharing your storage and it doesn't matter who is the "owner" of the storage.

It backs up to iCloud the same way as it would if she wasn’t sharing iCloud storage; i.e., she continues to access iCloud in the same way she did before sharing storage. The only change is the amount of storage available and the billing. Also see: How to back up your iPhone, iPad, and iPod touch with iCloud - Apple Support and Share iCloud+ with your family - Apple Support