What do I do when my icloud is different than my Apple ID and I want to back up?

An iCloud account is defined by the AppleID used to create it and to log into it. So if you each have different iCloud accounts then you also have different AppleIDs even if those are only used with iCloud. You can each backup your device(s) to your iCloud account.

You both really should be using your own unique AppleID for everything. Set up family sharing for sharing purchased content and or an iCloud storage subscription.

Family Sharing - Apple

You can backup to iCloud or to a computer. How to back up your iPhone, iPad, and iPod touch - Apple Support

Each user should have their own Apple ID.

Families and Kids - Official Apple Support