You are getting the same network card in both browsers because you have authenticated same user via GitHub in both browsers.
The github authentication scheme is a 'delegated' authentication scheme and is using a client application (id and secret) authentication mechanism to enable a client to use the REST APIs and get a token, without needing to authenticate as a user every time.
Once a client has authenticated to the REST API, that client can add Blockchain identities to a wallet. The wallet is private to that client, and is not accessible to other clients. When a REST client makes a request to the REST server, a Blockchain identity in each clients wallet is used to digitally sign all transactions made by that client (because the identity is set as default, at the time).