It is generally accepted that when someone uses the phase "stateless API" they mean session state not data persistence.
Yes, and this does nothing to help with the problem of state coherence between the backend and frontend.
Most JSON/data APIs for SPAs are similar to SQL database servers: maintaining a persistent connection is useful for performance reasons, but there's nothing about the API that dictates that it's required. The only state that is required across API calls is an authenticated session, and that is maintained on the server side.
and even authentication can be done stateless. Makes load balancing easy since each request can go to a different server