More specifically, if I go from dropped C tuning to dropped B tuning, will the same C minor scale that i've memorized in dropped C be in the same position in dropped B? Or would be be B minor?.
It would be B minor.
There's no such thing; there never was. Where I am going you cannot follow me now.
Doesn't matter what instrument you are using or which position you are playing in. It's the notes you play that matter.

If you use the notes C D Eb F G Ab and Bb, you are playing in C minor (again, no matter what instrument you use or what position you use). When you change your tuning, the position of the notes changes. But C minor is always C D Eb F G Ab Bb.

And yeah, if you play in the same position and change your tuning from C to B, it will be B minor.
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