You are searching the JIRA custom field your Administrator has configured. You are not searching the user profile fields directly. Therefore you have to use the name of the custom field for the search. The name might differ from the name of the user profile field.
The value of the custom field is always relative to a reference user field in the ticket. You might get results based on the user profile of the Assignee, the Reporter etc. This too depends on the configuration made by your administrator.
If the search doesn't work as expected, please ask your administrator for details on the configuration so you are aware of the current set-up and how it is supposed to work.
You can find an example below which illustrates the behavior of the search.
Configuration: The user profile field Manager has been mapped to the JIRA custom field Manager of Reporter. As the name suggests, the field Manager of Reporter is set up to display the manager of the reporter in an issue (provided the user profile field Manager has a value).
Image 1: Relationship between user profile field and JIRA custom field
The manager of the current user is Jane. Therefore if you open an issue where the current user is set as Reporter the field Manager of Reporter will display the value "Jane".
To search for issues with the Manager of Reporter Jane
click "More" and search for the field Manager of Reporter
after selecting the field, enter the value "Jane"
You will now see all tickets where the field Manager of Reporter is filled with the value "Jane". This will include tickets where the current user is set as reporter, but it can e.g. also include tickets where colleagues of the current user who also report to Jane are set as reporter.
You can now refine your search by adding values for other JIRA fields, e.g. filtering for a specific project or for tickets with a specific reporter.