I believe if I book online and travelling to another Schengen country, I would not need to show ID and only the boarding ticket?
That depends on the airline's policy (and to some degree, the gate agent's knowledge of that policy).
The EU no longer requires that airlines check for their passengers' identification so long as they do not check in luggage. Many airlines (SAS, Norwegian) do not care and do not check in line with regulation. Ryanair, however, systematically checks.
I regularly fly between Sweden where I live, the Netherlands, Germany, Spain, and Italy and I never have to show my ID. But if the airline does ask...
However, as a whole, being in a different country without an ID is not recommended and as Alexander points out, some countries such as Spain and France do require you carry some form of identification at all times e.g. national id card or driver's license