[FEATURE] Projects grouping in the desktop Code tab (asymmetric with Home tab today)

Preflight Checklist

• [x] I have searched existing requests and this specific angle (desktop Code tab, matching the Home tab paradigm) hasn't been filed
• [x] This is a single feature request (not multiple features)

Problem Statement

Companion to #75126 (same idea, platform:vscode labeled), #83279 (bulk-import angle, desktop-flavoured), #74129 (nested-folder tree in Code tab), and #73802 (Home Projects → Code sync). This issue focuses on the desktop-app Code tab specifically.

Per code.claude.com/docs/en/desktop, the desktop Code tab supports filtering by Home projects but has no way to create or organize Projects in the Code tab itself. The Home tab has Projects; the Code tab does not. That asymmetry within the same desktop app is the problem.

Users working across multiple codebases in Claude Code Desktop end up with a flat list of sessions in the Code tab sidebar, no grouping, and no way to say "these sessions belong to this project" without reading each title. Every user re-invents the pattern either mentally or via out-of-app notes.

Proposed Solution

Add Projects grouping to the desktop Code tab, matching the paradigm the Home tab already has:

1. Sidebar section listing Projects, one per directory the user has ever launched Code from.
2. Auto-populate from ~/.claude/projects/<encoded-cwd>/ — the state is already on disk; this is a UI-surfacing change, not new persistence.
3. Human-readable name derived from the last path segment; overridable via ~/.claude/projects/<cwd>/project.json with a displayName field.
4. Clicking a Project filters the session list to that project.
5. When a session is active, the sidebar highlights which Project it belongs to.
6. Session count + last-activity date per Project.

Alternative Solutions

• Read each session title to guess project membership.
• Keep an out-of-app note mapping titles to projects.
• Restrict work to one project at a time.

None scale past a couple of projects.

Priority

Medium - Would be very helpful

Feature Category

Other

(Suggested label if applicable: area:desktop. The Category dropdown has no clean fit for a desktop-app-UI request; noting here.)

Use Case Example

As a user, I work across multiple codebases in Claude Code Desktop. Today the Code tab sidebar shows a flat list of ~30 session titles from every folder I have ever launched Code in. To continue work on a specific project, I have to read each title to guess relevance. With Projects grouping in the Code tab (matching the Home tab), I click the project's name and see only its sessions, most recent at top, and resume in one click.

What's new vs the related issues

• Desktop Code tab surface, not the VS Code extension of #75126 and not the CLI. No open issue targets this surface specifically for the Projects concept.
• Symmetry with Home tab — bring the paradigm across within the same desktop app, rather than reinventing.
• Auto-populate from ~/.claude/projects/ — no other issue observes that the state is already there on disk; this is UI surfacing, not new persistence.
• project.json displayName override — small addition on top.

Additional Context

• Upvoted #75126, #83279, #74129, #73802 as related.
• Prior related closed as duplicate: #23017.
• project.json shape suggestion: { "displayName": "OneDay Insight" }.