Text messages sent to my wife's iPhone are being received on my iPhone with the same Apple ID

I think you already know the "fix" for this.......each person needs to have their own Apple ID.

But, you can try the following to see if it might help things as they stand now:

On your phone

Tap Settings

Scroll down and tap on Appls

Tap Messages

Tap on Send & Receive

Make sure that the check mark only appears next your your phone number

Back out and come back to the Home screen

On your wife's phone

Tap Settings

Scroll down and tap on Apps

Tap Messages

Tap Send & Receive

Make sure that the only check mark appears next to your wife's phone number

Back out and come back to the Home screen

No guarantees, but it's worth a try.

There is no problem here, as this is all working entirely as designed and intended.

With one person active, the messages and two-factor notifications and therest will be shared across all, and will arrive on the active devices.

If you want separate environments, thrn create separate Apple Accounts.

Make sure that each family member has a unique Apple Account - Apple Support

Once everybody has their own accounts, and usually with Family Sharing enabled, here is how to migrate data:

How to “unshare” an Apple ID - Apple Community