As has been mentioned above, you can have a custom basemap gallery in Portal as you can in AGOL and that is the correct documentation for configuration.
The main steps are as follows - create a new Custom Basemap Gallery Group in Portal, add the basemaps that you would like to be visible and then set this new group as the Default Basemap Group in the Portal's Organisation page (this will require being an Administrator or a role with equivalent privileges).
This allows you to share the following as custom basemaps:
- Existing basemaps (such as the global basemaps from the default basemap gallery)
- Web maps