[FEATURE] Don't anonymize user-assigned browser names ("Browser 1", "Browser 2") sent to the model

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

When connecting Claude in Chrome, the extension lets me assign a name to each browser, stores it, and re-proposes it on later connections (e.g. "Laptop (Edge)", "Desktop (Brave)"). However, the data passed to the model via list_connected_browsers replaces these names with generic positional labels: name: "Browser 1", name: "Browser 2". Only the deviceId (a raw UUID) survives.

This breaks the feature in a way that hurts the user it's supposedly protecting:

• The name I authored is hidden, and I'm handed a UUID that is meaningless to a human. When the model shows the browser picker, I cannot tell which entry is which machine.
• For anyone running more than one connected browser/machine, this is a real footgun: selecting the wrong entry means sending browser actions to the wrong computer.
• The "privacy" justification is hard to follow: the name is user-authored data, stored by the extension and shown back to me anyway. Masking it from the model doesn't protect me from myself; it just makes selection unreliable.

Impact: multi-browser / multi-machine users can no longer reliably identify the target browser. The picker degrades from "choose by name" to "guess by UUID and ordering".

Proposed Solution

Pass the user-assigned display name through to the model (or expose an opt-in setting), so list_connected_browsers returns the real label alongside the deviceId. At minimum, make the anonymization toggleable per account.

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_