I am planning to do something similar to Session sharing between two application deployed on same Tomcat Server though I am using VueJS rather than Angular.
In Tomcat 7+ settings there's a way of setting what the possible values are for session management. My app presently uses cookies, but assuming I can pass the session ID using HttpSession.getId() to a Javascript, likely embedding in the generated JSP which bootstraps the VueJS code or passing it as a query parameter.
The question is whether that approach would work. I'll be finding out anyway because I'm trying to set up a PoC to do this, but in case anyone knows off hand at least it would save me time for experimentation.