How can I find out if my display is genuine and how do I stop the notification from popping up

Did you buy the phone used? If you ever had any repairs done, were they done by Apple or an Authorized Apple Service Provider? If the answer to either of those questions is "Yes", the display may not be genuine. If the answer is "No", there may be something wrong with it.

The pop up notification will stop. However, there will continue to be a notifications in Settings>General>About>Parts and Service History. It will not interfere with your use of the phone. The only way to remove that is by having the phone properly service by Apple or an Authorize Apple Service Provider.