Your use case is precisely what a stateful session bean is for, if you want your List object to be maintained across method invocations, and if you want each session to be assigned its own bean.
Stateless session beans are pooled and made available to any session. But your instance fields are not guaranteed to be cleared, so you can't depend on them being cleared. The behavior that you are seeing is not unexpected. Even if you were successful in creating separate sessions in multiple tabs, those sessions could very well have been (and apparently were) assigned the same session bean. That's because the associated method invocations occurred at different points in time. Now if the associated method invocations occurred simultaneously instead, then the platform would have assigned a different stateless bean to each invocation (session). In that case, you'd see different behavior.
See also;
conversational state of session beans
and
Stateless and Stateful Enterprise Java Beans
Never let what you can't do get in the way of what you can do.