ArcGIS Enterprise 11.0\ Pro 3.0.1: What is the function of the “feature” option when publishing a data with “reference registered data” (map image) option?
It is a service capability. When you share as a referenced service, Map is enabled as a minimum, but you can also enable Feature capability i.e. a feature service is a map service with feature access enabled.
This allows for more functionality against the reference data i.e. you will need feature access enabled (and add/update/delete) for editing the data.
But how this option is available in case of file geodatabase? Editable map service (feature service) requires layers that are stored in enterprise geodatabase. Am I correct?
as @magisian said Feature is only to enable Feature Service (aka feature access) capability off a map service to allow editing from web clients.
By ref feature service doesn't support filegdb therefore I believe you should get an analyzer error if you try to publish it from ArcGIS Pro. Yes, you are right the map must only have data from enterprise databases as you mentioned above.
Please note (for folks who are not aware of), if you just want to add a layer off a map service as a feature layer (not a map image layer) in your web map, you can still do that off a map service and the feature option is NOT needed. All you need to do is to use the url of the layer within your map service (instead of map service's root url) while adding a layer in your web map.
Here is an example of a url that will add a feature layer (not a map image layer): https://sampleserver6.arcgisonline.com/arcgis/rest/services/SampleWorldCities/MapServer/0