After playing around with this a bit in my dev org, this occurs when no customizations have been made to the Lookup Phone Dialogs on the Search Layouts of an object and custom phone numbers are introduced. Steps to repeat experiment:
On User Object with the default Search Layouts Lookup Phone Dialog intact, create a new custom field User.Other_Phone__c. If you look at the Lookup Phone Dialog Layout you will notice the field was automagically added. You will also notice the field Id is pulled down in the metadata vs the api name.
Manually relabel the Id to the API name and deploy, then refresh metadata from server and API name sticks.
Just to ensure metadata changes and API name is used, modified the order in the UI of the phone fields and refreshed again from server.
Added another custom phone field Another_Phone__c and refreshed from server. API name was automatically used. Experimented on the contact object and modified the layout first before adding the custom field and the API name was used vs the Id.
A report of this issue can be found back in 2013 so it does not appear to be a new issue but in my mind it is a bug or else there should be mention of it in the Metadata API Developer's Guide. Currently states the following:
The list of fields displayed in a lookup dialog for the object. The
name field is mandatory and is always displayed as the first column
header, so it is not included in this list; all additional fields are
included. The field name relative to the object name, for example
MyCustomField__c, is specified for each custom field.
Salesforce objects often include one or more lookup fields that allow
users to associate two records together in a relationship. For
example, a contact record includes an Account lookup field that
represents the relationship between the contact and the organization
with which the contact is associated. A lookup search dialog helps you
search for the record associated with the one being edited. Lookup
filter fields allow you to filter your lookup search by a customized
list of fields in the object.
This field is equivalent to the Lookup Dialogs in the Search Layouts
related list on the object detail page in the application user
interface. For more information, see “Lookup Dialog Search” in the
Salesforce online help.