I think cookies would still be used to identify session on server side.
Otherwise you'd have to extract SessionID from sessionStorage, then somehow send it (GET/POST/HEADER) each time user accesses resource where session identification is required.
Isn't it so that sessionStorage does not send data to the server. I thought you had to write your own script to send info to the server...
Yes - that is the point.
But how will server know, which user/session to serve?
an ajax call doesn't use session cookies
Yes, you could do without cookies, but is it worth it? You'd have to reimplement whole session management mechanism (setting/transmitting/destroying session token) on client side.