Eb dorian will share the same key signature as Db Ionian (not B). The difference is the tonic, or key center. Just play the standard Db major (Ionian) scale. Then play the same scale, except start and end on Eb instead of Db. You'll get a different feel. It has a minor touch.
An example you're probably more familiar with is C Ionian/A Aeolian. If you play from C to C with no sharps and flats, you get a standard Ionian major scale. If you play from A to A with no sharps or flats, you get a totally different sound. You now have a minor (Aeolian) scale. Yet, the key signature is the same in both examples. By changing where you center the music, you can create different moods.