A person may assert several simultaneous gender identities, for example in different settings. They may also use different names in those different settings.

Is there a recommended way to indicate which GI and which name belong together?

If so, what is it?

Is this just a theoretical problem?

What are some real-world examples?

The relationship between the concepts could be
1/
name is used in setting and
gender is used in setting
This introduces a new concept of setting or setting type
2/
name is used with gender
In FHIR this could be an extension on name. The extension could be a gender element or a pointer to an existing gender element elsewhere in the resource.
3/
something else

See also (556) #Gender Harmony > Gender identity and name - FHIR Community - Zulip