[FEATURE] Reuse existing Chrome tab group instead of creating new ones

Problem

When using Claude in Chrome, a new tab group is created for each task. This leads to accumulating tab groups even when an existing Claude tab group is already available and could be reused.

Current Behavior

• Start a task → Creates new "Claude" tab group
• Start another task → Creates another new tab group
• Tab groups accumulate over time

Expected Behavior

If a Claude tab group already exists, new tabs should be added to that existing group instead of creating a new one.

Related Issues

• #15436 addresses tab group accumulation via cleanup after sessions end
• This issue proposes preventing accumulation in the first place by reusing existing groups

Environment

• macOS
• Claude Code + Claude for Chrome (Beta)