Implement ID-Based Approach for Tracking Cloned vFolders

## Description

Currently, the vfolder structure in [Backend.AI](http://Backend.AI) does not allow users to easily identify the origin of a cloned folder by its name, as folder names can be changed. To improve usability and management, it is suggested to implement an ID-based method to track the origins of cloned folders.

## Acceptance Criteria

- Develop a new table to track the relationship between original folders and their clones using an ID-based approach.
- Implement an API (preferably GraphQL) that can:
- Determine which folder a cloned folder originated from.
- Check if the current user owns the cloned folder.
- Support filtering capabilities to facilitate ownership checks.
- The new table should establish relationships between folders to indicate cloning origins but should not be used to restrict cloning operations at the server level. This is to allow future cloning operations if needed.
- The client should be able to display warning messages based on the cloning relationships indicated by the new table, without server-level restrictions on cloning.

## Rationale

This enhancement aims to improve folder management and user experience by providing clear visibility on the origin of cloned folders. An ID-based tracking system will allow for more precise and efficient management of folder relationships and ownership.

## Implementation Notes

- The API's design should prioritize scalability and efficiency, considering the potential growth in the number of vfolders and clones.
- Careful consideration is needed to ensure that the implementation does not introduce significant performance overheads, especially in environments with a large number of folders.

[View original message posted by Sujin Kim on 7/25/2025, 1:38:10 PM](https://teams.microsoft.com/l/message/19:3c0acc0825024daeb2211f55c4e7f4aa@thread.skype/1753418290112?parentMessageId=1753418290112)

JIRA Issue: BA-1975