How can I login with same user in multiple browser, API same time?
If you look at this post you will see everything explained.
In order to summarize, what it is telling you is that by default you just can't log in Sugar with the same user using using two or more different browsers at the same time. This is because you are trying to use the same platform (which is the base one) in order to log in to Sugar. However you can for instance enter with the same user in two different mobile devices as the mobile platform allows a maximum of two sessions at the same time with the same user.
Moreover if you want to be loged in Sugar with one user and be able to perform REST API calls with the same user at the same time you only have to specify a different platform in your call as explained in this article. For instance I always use platform = api when making REST API calls.
These restrictions are in order not to let Sugar users to share their credentials and utilize the same user between several people.
Nevertheless, you can still override this default configuration and specify a maximum number of concurrent sessions per user. You just have to override the platform specific configuration files. For instance if you want to be able to log in two different browsers with the same users at the same time you would override the file include/SugarOAuth2/SugarOAuth2StorageBase.php which has the attribute numSessions = 1 in folder custom/include/SugarOAuth2/SugarOAuth2StorageBase.php and put there your new configuration.
So please, do not override this if your purpose is to let several people to log in with the same user as Sugar licenses are paid per user!