Text messages wife sends appear with my name

That is odd if you both have your own Apple ID.

One place you could check on her iPhone would be:

Settings > Messages > Send & Receive > check to make sure only her phone number and/or email address is checked

Check your contacts, you probably have a contact card with both of your names on it.

If she looks in Contacts on her phone, does the word “me” appear against her own contact on RH side?