Family Sharing and iMessage ID

Recently, my wife and I decided we both wanted separate medical IDs on our Apple devices. We realized that if we were both on the same Apple ID, there could only be one medical ID. So, I decided to create a new Apple ID for my phone so that I could have my own medical ID. Then she invited me to our family via Family Sharing. I was a member of the family with parent/guardian status. So far so good.

However, whenever I send a message from my iPhone to hers, the drop-down alert on her iPhone identifies me as the sender, but when she taps on the message to open it, the opened message shows my wife as being the sender.

Can anyone advise me how I can deal with this issue?