Be careful that you are using RxJS v5 while your documentation link seem to be RxJS v4. I don’t remember specifics but I think that the share operator went through some changes, in particular when it comes to completion and resubscription, but don’t take my word for it.
Back to your question, as you have shown in your study, your expectations do not correspond to the library design. Observables lazily instantiate their data flow, concretely initiating the dataflow when a subscriber subscribes. When a second subscriber subscribes to the same observable, another new dataflow is started as if it is was the first subscriber (so yes, each subscription creates a new chain of observables as you said). This is what is coined in RxJS terminology as a cold observable and that’s the default behaviour for RxJS observable. If you want an observable which sends its data to the subscribers it has at the moment the data arrives, this is coined a hot observable, and one way to get a hot observable is to use the share operator.
You can find illustrated subscription and data flows here : Hot and Cold observables : are there ‘hot’ and ‘cold’ operators? (this is valid for RxJS v4, but most of it is valid for v5).