Both phones use the same apple ID. How do I change it on 1 phone?

I hope I got your situation right:

Do either you or your wife have a second Apple ID? You're getting the error because, if you created a family with your account, you can't send you wife an invitation to it because "she is you".

Create a different account for at least one of you (or switch to an existing other one), set that person's phone up using that alternative Apple ID and then send an invitation to the other person addressed at their new Apple ID. That is: one of you keeps the account you have shared so far, the other one created/switches to a new one. Then you set up Family Sharing between the two of you.

Family sharing is designed to share content between different apple Id's. Not between different devices with he same ID.

If she is using your ID, then you are technically inviting yourself. which is pointless, as that Apple Id already has access to its own content.

You can go to http://appleid.apple.com and Create the new Apple ID for her.

Then sign out of the relevant services on her phone and sign in with the new Apple ID.

On her Phone:

Settings->iTunes and App Sores->Apple ID->Sign Out

Settings->Facetime->Apple ID->Sign Out

Settings->Messages->Send & Receive->Apple ID

Settings->iCloud->Sign Out

Though If you are using one Id and you have no issues with sharing one ID and all the content, I see no need to create a second ID just for a sharing feature you technically do not need.

Family sharing is geared more towards people with different ID's that want to share purchases, calendars, and other content form a single common ID, while still keeping access to their own Apple ID's and its contents and purchases.

Create an Apple ID: https://appleid.apple.com/account

2.) No, you do not need to restore the phone. Just go to Settings>iCloud as well as Settings>iTunes and App Store, tap on the account name, sign out, then sign in with the new one. Also do this with iMessage and Face Time as well (in Settings, again). Be sure to transfer any documents in iCloud (if any) etc. that she might want to hold on to in her new account to that new account just by mailing them to her (either before or after setting up the phone). Music she need not worry about, as purchases are synced across a family, once it is created. Same for photos. Do continue to sync the iPhones to the same iTunes library however (if you already do so) so that she retain access to imported CD music and the likes. There might be some inconvenience over the next few weeks regarding Auto-Updating apps because they were bought on the old account and Apple doesn't transfer purchases,mso she'll have to manually re-download them with her new account, though she won't be charged of course, as long as she's part of the family. I think I didn't forget anything.