Yes, Neighborhood is there in the Address Fields, but that is for multiline input, and that doesn't mean you can search on them from the Search widget. There are 2 input types for geocoding: multiline and singleline. The Search widget only uses singleline, and suggestions only work with singleline anyway.
For this to work with the Search widget, you would need to publish or configure a custom locator service. I would suggest that you use the Create Locator tool in ArcGIS Pro with the “Neighborhood” role. Here is more information on that process:
https://pro.arcgis.com/en/pro-app/latest/help/data/geocoding/introduction-to-custom-locators.htm
You could also use the locator class and build your own UI, instead of using the Search widget. Here is a tutorial that walks you through the basics:
https://developers.arcgis.com/javascript/latest/find-places/
Unfortunately, I think these are your available options. There might be a clever way to overwrite the getResults() and getSuggestions() methods, but I don't know for sure if this would work.
https://developers.arcgis.com/javascript/latest/api-reference/esri-widgets-Search-LocatorSearchSourc...