The modes can indeed be said to be the same scale but starting out in different points, but you've missed a crucial point: If you play a C major scale starting on D, you are playing a D dorian, not a C dorian. C dorian is the second mode of Bb Ionian, and the notes are: C, D, Eb, F, G, A, Bb.
Now, you could say that when you played a solo in C major, you were playing in the modes C Ionian, D Dorian, E Phrygian, F Lydian, etc. which would be more correct than saying you were playing in C Ionian, C Dorian, etc (which is wrong, as I note above), but it would still be not correct.
As TheDirt says, the mode is related to the tonal centre. So, when you play over a C major scale you are playing in C Ionian (or C Lydian maybe), because the tonal centre is C. YOu are not playing in E Phrygian, because the solo doesn't resolve to the E.
Does this help a bit?
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