Support GitHub App authentication across multiple organizations/installation IDs

Issue #1333 tracks a request to enable GitHub App authentication as an alternative to current options like PAT or OAuth.

This issue tracks a further request that it be possible to configure the GitHub MCP Server to authenticate with GitHub App authentication across multiple installation IDs/organizations. In some scenarios, users may have repositories spread across multiple organizations in their enterprise. Although GitHub Apps can be installed in multiple organizations, each org will have its own installation ID and will need to have a separate access token minted to authenticate with it.

Existing proposed solutions to #1333 (such as #2797) enable the GH MCP Server to authenticate with GitHub App Auth but only for a single installation ID. It would be great to have a more robust solution that allowed configuring the MCP server with an App ID and private key file that could be used across multiple organizations/installation IDs.

Start with issue #1333 and the proposed single-installation solution in #2797 to understand the existing GitHub App authentication scope. Trace how an App ID and private key would be used across installation IDs and organizations; done means the server can authenticate to repositories in multiple organizations with separately minted installation tokens.

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