I want to use my way to log in via Java Action by calling com.mendix.core.Core.login(Username, Password) which gives me a ISession object. However, the user in the browser remains an anonymous user. How can I change the session of the current user in the browser through a Microflow or Java action?
You login in the runtime, but the browser is not aware of that.
You need to set session cookies etc. See login widget for details. Also the Single Sign On module is creating that cookies.