Claude in Chrome: multiple connected browsers force an unanswerable per-session picker; need per-project browser profile bindings

Problem

When the Claude in Chrome extension is installed in multiple Chrome profiles on the same machine, list_connected_browsers returns opaque entries ("Browser 1", "Browser 2", "Browser 3") with no identifying information (no Chrome profile name, no signed-in account). The tool result then instructs the model that it MUST stop and ask the user to pick a browser (via AskUserQuestion) before ANY browser action, every single session.

For users who keep separate Chrome profiles per company/project (a very common setup), this means:

1. Every session that touches the browser begins with a blocking question the user cannot meaningfully answer - "Browser 1/2/3" carries zero information about which profile is which.
2. Autonomous/long-running sessions stall at the very first browser step.
3. There is no way to persist the choice - the same question fires again in the next session even though the answer never changes for a given project.

Feature requests

1. Surface identity in list_connected_browsers: include the Chrome profile name (and ideally the signed-in browser account) so the choice is meaningful.
2. Persistent browser naming: let the user name each connected browser once (the switch_browser connect flow already allows this) and keep that name stable across sessions.
3. Per-project default binding: allow a project-level setting (e.g. chromeDeviceId in .claude/settings.json, or "remember this choice for this project" on the picker) so each project is bound to one browser profile and the picker is skipped when a binding exists.
4. Honor the binding in select_browser: when a stored binding matches a connected browser, select_browser should proceed without the mandatory ask-the-user step.

Workaround currently used

Probing each browser to identify the profile, then storing the deviceId in the user's global CLAUDE.md with an instruction to select_browser directly. This works but is fragile (deviceIds are opaque and can change if the extension is reinstalled) and burns tokens/time on probing.

Environment

• Claude Code CLI on macOS (Darwin 25.5)
• Claude in Chrome MCP (mcp__claude-in-chrome__* tools)
• 3 Chrome profiles with the extension installed, all connected locally

🤖 Filed with Claude Code on behalf of the user