[FEATURE] Desktop Code tab: user-defined nested folder hierarchy in the session sidebar (Obsidian-style)

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

The Code-tab sidebar only filters by status/project/environment and groups by project — effectively a single level. I manage many long-lived sessions across research areas and projects, and one flat grouping level isn't enough to keep an overview.

I can't create nested folders (e.g. Area > Project > Subproject > session), define my own folders, or drag sessions between them, the way Obsidian's file explorer lets me organize notes into a deep tree.

Proposed Solution

• User-definable nested folders in the sidebar, arbitrarily deep and collapsible.
• Drag-and-drop of sessions (and folders) to organize them.
• Persist the tree across restarts. Ideally pair it with a tabbed-session model (#74127) so I can browse the hierarchy on the left while working across tabs.

Alternative Solutions

Today only "group by project" (one level) exists. Related existing requests: #68262 (subfolders / nested organization for Claude.ai Projects — related but broader / different surface) and #22617 (closed: folders + pinning for session organization). Obsidian's nested folders are the model I'd want in the Code-tab sidebar.

(Companion requests from the same workflow: #74124 visit-order Back/Forward navigation, #74127 tab-based session UI.)

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. Under a top folder "Org A" I keep "Center X > Project 1 > session" and "Center X > Project 2 > session".
2. A separate "Personal" tree holds unrelated sessions.
3. I collapse whole branches I'm not using, instead of scrolling one long flat list grouped only by project.
4. I drag a session into a different subfolder when its scope changes.

Additional Context

Platform: macOS (Apple Silicon), Claude Desktop, Code tab. Current sidebar grouping is documented at https://code.claude.com/docs/en/desktop (filter by status/project/environment, group by project — one level).

(Feature Category is set to "Other" because the form has no Desktop-app / UI category; this is a Desktop Code-tab UI request.)