Another thing, border control is removed between countries in Schengen.
It's a common myth, but it's not true.
What happens with Schengen is that the static checks on country borders are replaced with a 'hard' outer shell and a 'soft' inner shell. The shell still exists however, and is manifested usually with random border checks away from the border. The execution is different in different countries - for instance, France will check you anywhere at any time, whereas Austria is known for pulling people over 500m from the border.
For that reason, it's vital for ID to be carried at all times when in the zone.