You can run any two routers with the same name and password as you please. I do not believe that your client would automatically connect to the second router if it originally signed in with the first. As the wifi standard only has the password sent to the router the first time it is used, after that a session token is used. So let's imagine that you logged in with your phone to the first router, a session key was generated, everything was working fine. Then you walked over to the next room and your phone attempted to connect to the second router. Well that connection would be denied as the second router will not have the session key stored on it so it will not be able to authenticate your phone as having properly logged in. This is why large businesses and schools use enterprise grade products offered by large companies like Cisco. It's because they offer software and servers that allow for the sharing of session keys across routers. Now I'm not saying that you wouldn't be able to set something up like this yourself I'm simply saying that you won't find much help online as most people aren't using the Pi and that it is more difficult than you think.
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