Feature: populate session_scope.displayName so Claude-in-Chrome tab groups are labeled per session

Problem

When several Claude Code CLI sessions run in parallel (e.g. one session per ticket) and each uses Claude in Chrome, every session gets its own Chrome tab group, but all of the groups are titled "Claude" and differ only by color. With 4+ concurrent sessions, often holding visually identical tabs of the same local app, there is no way to tell which tab group belongs to which CLI session, in either direction.

The extension already supports the fix

Looking at the installed extension (v1.0.85) service worker, tab-group naming is already implemented and driven by the session_scope object received over native messaging:

• On group creation: chrome.tabGroups.update(groupId, { title: sessionScope.displayName ?? "Claude", color: ... })
• On later context fetches, guarded by displayName &&, the title is re-asserted when a displayName is present.

Claude Desktop populates displayName (which is why Desktop projects get named tab groups). The CLI's session_scope carries only the session ID and tab group ID, with no displayName, so CLI-created groups always fall back to "Claude".

Request

Have the CLI populate session_scope.displayName so each session's tab group is identifiable. Any of these would work, in order of preference:

1. A user-controllable value: a setting or env var (or reuse the terminal/session title if one is set).
2. A sensible automatic default, e.g. the cwd basename or the git branch name, both of which usually encode the ticket in per-ticket-worktree workflows.
3. Even just the short session ID would beat four groups all named "Claude".

Since the extension side already ships, this appears to be a small CLI-side change with no extension update required.

Related issues

• #15193 (multi-session tab group isolation, open)
• #55332 (proposed passing session identity in session_scope, closed as duplicate)
• #17536 (renamable sessions generally)

None of the open ones ask specifically for the tab group label, so filing this separately.

Environment

• Claude Code v2.1.229
• Claude in Chrome extension v1.0.85
• macOS 15 (Darwin 25.6.0), Chrome stable