My phone and my wife's phone get the same text messages. How do I remove my wife's phone?

You need to use different Apple IDs for imessage and cloud on both phones. You can use the sameIDs for itunes store. also in imessage settings in send & receive make sure you do not have your wifes email address or number ticked and vice versa. You may need to close imessage on the phones reboot and then reset imessage

To be more specific I could have said

To fix this go to Settings>Messages>Send & Receive and do one of the following:

1. 1. Either uncheck the email address shown under "You can be reached by iMessage at" on both phones; or
2. 2. If you want to continue to receive iMessages at this email address on one of your phones, on the other phone tap the Apple ID at the top, sign out, then sign back in using a different Apple ID (so both phones will be using a different Apple ID for iMessage). Note: you can continue to share an Apple ID for purchasing from the iTunes and App stores. It does not need to be the same as the ID you are using for iMessage, FaceTime and other services.

Because you are both using the same Apple ID for iMessage. To prevent this, on one of the phones go to Settings>Messages>Send & Receive, tap the Apple ID, sign out, then sign in with a different Apple ID. Note: you can continue to share an Apple ID for purchasing; it does not need to be the same as the ID you use for iMessage, FaceTime, iCloud and other services.

Also, to prevent merging your calendars and contacts if you are syncing these to the same iCloud account, be sure to set up separate iCloud accounts on each phone with separated IDs.

While separate IDs is a good idea you do not need different Apple IDs to have each of you receive only your own text messages.

Go to Settings > Mesages > Send and Receive From

Deselect all but the phone number. Do this on both phones.