Wallet name is a shorthand term for "the name that appears on your government-issued ID."
Policies that require people to use their wallet names can lead to discrimination against certain classes of people, including Transgender people, people who don't have WASP names, etc. For a more complete list of people potentially affected, see Who is harmed by a "Real Names" policy?
Beyond that, it is false to assume that people's wallet names follow a certain pattern or will look a certain way or will solve certain problems. Some people are mononymous and have that on their government issued ID; some people have names with more than two parts; some names are very long or very short; and of course wallet names are not guaranteed to be unique in anyway. See also: Names are complicated.
Official name changes
In some countries (but not all), it is possible to change your wallet name through an official process. This process can be time consuming and costly, and is not required in Common Law countries (eg. the US, Canada, Australia) where you may effectively change your Legal name simply by using a new one.
This wiki documents several incidents where people have been required to use their wallet name, or asked to show government ID to use a service or participate in a community. These include: