Agent Window MCP server List does not match what is in VS Code

Does this issue occur when all extensions are disabled?: Yes/No

- VS Code Version:
- OS Version:

Steps to Reproduce:

1. Install Extensions within VS Code for example "Microsoft Fabric MCP Server"

2. Open VS Code Agent Window, then click on MCP Servers, This list does not match what I see inside VS Code

3. Remove the Fabric MCP Server from VS Code

4. Browse Marketplace inside Agent Window

5. Find MCP Server, and click Install

6. Click Install and now see MCP server inside Agent Mode

So issue, If you install MCP server via VS Code then go into Agent mode the MCP server will not show, but if you go back to VS Code remove the Extension MCP server, then the go back to Agent window, and reinstall then the MCP server will show.

Start by reproducing the mismatch through the Agent Window's MCP Servers list, comparing an MCP server installed in VS Code with one installed through the Agent Window Marketplace. Trace the Agent Window and MCP server installation entry points, then verify that installing or removing the extension through either path keeps the MCP server list consistent.

Written by the indexing model from the issue text.