Is it possible to detect file uploading/deleting in File Storage(not Blob Storage) and call some custom handler after that
As Wayne Yang -MSFT mentioned that azure file trigger is not supported by Azure currently. But Azure team is planning for it, deails you could refer to this feedback.
If Azure blob storage is possible, you could use the azure event grid to detect to the blob changes and trigger the Webhook with the function to save name and path of this file in my database.