Display Name Field is Not Changing
If you have added the display name to your profile form and it is not changing when a user changes the text in the field and updates profile then you need to make sure you are not allowing users to add their first and last names on your site. This is because WordPress automatically makes the display name a user's first and last name.
So as soon as a user adds their first and last name (via the pre-defined first and last name fields) this will become the display name which will prevent a user from changing the text in the display name field. If a user has not entered a first or last name, they will have no issue changing the text in the display name field.