[FEATURE] Display active GitHub org/repo name in Claude Code Web session UI

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 Claude Code Web UI (claude.ai/code) doesn't show which GitHub org/repo a session is connected to. The session header shows an AI-generated task title, the bottom bar shows branch names, and the sidebar lists sessions by task description — but the actual org/repo never appears anywhere on individual sessions.

There is a project filter dropdown in the sidebar that lists connected repos, but it's a filtering mechanism, not a label — you have to filter down to a single repo to know which sessions belong to it. When set to "All projects" or when multiple repos are selected, individual sessions are indistinguishable.

When you have sessions open across multiple orgs or similarly-named repos, there's no way to tell them apart at a glance without filtering or opening each one and reading the conversation context. Branch names like main don't help.

Proposed Solution

Show the org/repo name somewhere persistent in the session UI. The most natural spots:

1. Near the session header — as a subtitle or badge below the task title
2. In the bottom bar — next to the existing branch info, e.g., acme-corp/webapp · main ← claude/fix-auth-549uF
3. In the sidebar session list — below each session title

The data already exists — users select a repo at session creation and the branch bar references it. This is just surfacing it.

Alternative Solutions

The sidebar has a project filter dropdown that can narrow sessions to a single repo, but this is a workaround, not a solution — it requires actively filtering rather than seeing the info at a glance. There are no CLI-style workarounds (hooks, shell title hacks) available in the web UI.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. I have sessions open across 3 repos: acme-corp/webapp, acme-corp/api, and personal/dotfiles
2. I switch to the sidebar to find my API session
3. All three sessions show AI-generated titles like "Fix authentication flow" and "Update config handling" — no repo name visible
4. I have to click into each session and read context to find the right one
5. With org/repo shown in the sidebar and header, I'd identify it instantly

Additional Context

Related issues (all CLI/terminal-focused, none for web UI):

• #30993
• #26063
• #7229