At present the "Show user identity" option under "User Policies" only permits the following fields to be shown as part of the user identity.

It would add some additional flexibility if custom user profile fields could be presented as an option.

For instance, our ID number field is filled with an institutionally unique identifier for each user, however this information is pretty much only internally used.

Students are issued with a Registration Number each time the start a programme of study. This is much more commonly used to identify students on a daily basis, but as a custom profile field it cannot be shown on some of the more useful locations, such as the list of submissions into a class, user selection lists (e.g. enrolment)

Implementation

New behaviour for users

• The admin setting lists all 'text' type custom fields that have a maximum length of 255 characters or less, so you can choose these fields. Updated description text warns users that the custom fields do not work everywhere. [See screenshot]
• I have modified the code so that three features work with these options: 'Browse list of users', the Participants page list of users, and the 'Enrol' option on the participants page.
• In other screens, behaviour will be as before with custom fields not shown.

Developer changes

• Custom fields can be specified in $CFG->showuseridentity as 'profile_field_<shortname>', e.g. 'profile_field_favouritecolour'.
• New API \core\user_fields can be used to create lists of user fields, and SQL query fragments for getting them (which use LEFT JOIN to get any custom fields). It covers identity fields (showuseridentity), fields required for user pictures, and fields required for full names.

Compatibility

• The corresponding existing functions are all deprecated: get_extra_user_fields(), get_extra_user_fields_sql(), get_user_field_name(), get_all_user_name_fields(), and user_picture::fields(). All the features from these functions are subsumed into the new API.
• All existing code that called the deprecated functions, or used the $CFG variable directly, has been modified to use the new API. (The API option to exclude custom profile fields has been used, so that this massive change could be made without significant code changes.)

Modifying existing code

• As noted I only made it work for three features because it's far too much effort (and testing) to fix everywhere.
• It is usually relatively easy to fix each individual feature; the three commits for these features are quite short.