A particular feature collection layer can be saved as its own Item from the Contents of a web map by using the Save Layer option in the layer properties menu. The Item becomes a reusable layer in Content and can be shared with others as well as added to multiple web maps. However, a feature collection is technically a form of a JSON file, and does not have the same structure as a hosted feature service.
It is not possible to export a feature collection item directly to a file geodatabase. However, a feature collection can be published as a hosted feature service and the hosted feature service can then be exported to a file geodatabase. The steps are described below: