Are all Schengen member states are also members of the European Union?
No, not all of them. Four Schengen states are not members of the EU: Iceland, Norway, Switzerland and Liecthenstein. Also the Schengen Area includes several micro-states which, while not signatories to the Schengen agreement, are de-facto members of the Schengen area, since immigration control is being effectively delegated to a Schengen member state. Examples include Monaco, San Marino and Vatican City. Andorra does not participate in the Schengen Area and does require a national ID card or passport in order to enter.
There is a different situation with respects the United Kingdom and Ireland, which are both EU members, but are not part of the Schengen Area - they instead implement their own 'Common Travel Area' for historic reasons. Normal border controls exist when entering the UK or Ireland from a Schengen state, however there are no controls when travelling between the UK and Ireland.