When initiating an OAuth authentication, it is possible to provide additional parameters, meant for custom use of the client application.
These parameters are correctly passed to your https://airtable.com/oauth2/v1/authorize page, as URL query parameters along the standard parameters (nonce, response_type, code_challenge_method, scope, code_challenge and client_id).
However when the redirect_url is called after a successful authentication, the additional parameters are not included. That would be a standard behavior your authentication should support.