[claude-in-chrome] MCP tab group always named "Claude" — parallel sessions are indistinguishable

Summary

The claude-in-chrome MCP creates/reuses a Chrome tab group that is always labeled "Claude" with the same status icons (checkmark / hourglass). When multiple Claude Code sessions run at once, every session's tab group looks identical, so there is no way to map a tab group back to the session that owns it.

Steps to reproduce

1. Start two (or more) Claude Code sessions, each using the claude-in-chrome MCP.
2. Each session calls tabs_context_mcp (createIfEmpty: true) and gets its own tab group.
3. Look at the Chrome tab strip: every group is named "Claude" with the same icon set.

Expected

Each session's tab group should be distinguishable, e.g. a group name derived from the session name/id or a configurable label, and/or a distinct group color per session.

Actual

• All groups share the identical name "Claude" + checkmark/hourglass status icons.
• Renaming the group manually in Chrome does not persist: the next time the agent operates on the group, it reverts to "Claude".
• The agent cannot fix this itself:
• chrome.tabGroups is not reachable from the page JS context (javascript_tool reports chrome.tabGroups is false).
• No MCP tool exposes tab-group rename or color.
• The computer tool only addresses the page viewport, not the browser tab strip, so it can't right-click the group header either.

Impact

Anyone running parallel sessions (a common workflow) cannot visually tell which tab group belongs to which session. Cosmetic, but it adds real friction during multi-session work.

Request

1. Allow a per-session tab-group name (session name / custom label) and/or color.
2. Stop overwriting a user-renamed group back to "Claude" on reuse.
3. Optionally expose a tabs_set_group_label MCP tool so the agent can set it.

Environment

• Claude Code (Opus 4.8), Windows 11
• claude-in-chrome MCP