Based on your comment, you are trying to remove the Name field from a Contact's Search layout. Below is an excerpt of what I see in the documentation. Though it does not specifically call out anything about Contact, but it does look like that this is applicable for Name fields across all objects as they uniquely identify a record.
You can’t remove unique identifying fields, such as Account Name or Case Number, from the search layouts. These fields must be listed first in the order of fields in the search layout.
You can verify this by trying to remove the Account Name field on Account search layout and you will get to see a similar error as you have reported.
I also verified this on a custom object, trying to remove the Name field, and received an error (not exactly the same because it was not related to person account at all), but it does support that Name fields cannot be removed from search layouts.
Error: abc Name cannot be removed and must be the first field in the search layout.