[FEATURE] Desktop Code tab: tab-based session UI within one window (Obsidian/browser-style tabs)

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

In the Desktop app's Code tab, open sessions appear only as a vertical sidebar list. To see more than one at a time I must Cmd+click to open a split pane, which doesn't scale when I want to keep many sessions open and flip between them.

There is no tab model: I can't keep, say, 8 sessions open as tabs and switch / reorder / close them the way I do in Obsidian or a web browser. Split panes are for viewing two at once, not for managing a large working set.

Proposed Solution

• Add a horizontal tab strip for open sessions: open many as tabs, switch by click or keyboard, reorder by drag, close individually.
• Keep it alongside the existing sidebar (sidebar = full library; tabs = currently-open working set).
• Optionally allow a tab to be torn off into an OS-level window.

Alternative Solutions

Workarounds today: Cmd+click split panes (only two at a time) or launching a second app instance via open -n -a "Claude" (doubles memory).

Related existing requests about extra windows: #30154, #27725, #60879 — related but distinct: those ask for multiple / detachable windows, whereas this asks for an in-window tab model (many sessions, one window). Obsidian and web browsers already solve this with tabs. (Companion request: #74124 — visit-order Back/Forward navigation.)

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. I keep sessions for 6 repos open all day.
2. With a tab strip I click across them instantly and reorder them by priority.
3. Closing one tab doesn't disturb the rest.
4. Today, each additional session I want to see means another split pane or another app instance — which doesn't scale.

Additional Context

Platform: macOS (Apple Silicon), Claude Desktop, Code tab. The CLI-comparison table in the docs already labels the sidebar "Sidebar tabs", but there is no actual horizontal tab strip UI.

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