Add per-message copy button to Claude Code web interface (claude.ai/code)

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

Title: Add per-message copy button to Claude Code web interface (claude.ai/code)
Body:
The standard Claude.ai chat interface has a copy-response button on each message. Claude Code’s web interface (claude.ai/code) does not.
In Claude Code, responses are rendered as segmented markdown blocks (code blocks, tables, paragraphs), and browser text selection stops at container boundaries — making it impossible to select a full long response by click-dragging. Shift+click helps but is awkward for multi-page technical responses.
Request: Add a “Copy response” icon per message, matching the behaviour in the main chat UI.
Context: Particularly impactful for long technical responses (audit reports, code reviews, multi-table outputs) which are common in a coding assistant.

Proposed Solution

Add per-message copy button to Claude Code web interface (claude.ai/code)

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_