My new instructor explained his reasoning a little better. He doesn't really have a plan for me to promote through his system like a regular white belt would. His system is close enough to what I already do that he feels like I am going to run through it rather quickly. I just need to learn the particulars of his system. So instead of having his white and yellow belt students wonder why I was working with blue belts when I was just with them a couple of weeks before, I should keep my current rank and then he can pair me up with whoever he wants at the time without causing confusion.
I think in arts that are fairly similar, there's usually no reason to start over at white, since most of your skills will carry over and its more an issue of picking up the new flavor more than learning new techniques etc. If the arts were signifigantly different (say, you moved into Judo) then I'd think it would be silly to wear a BB while you're essentially starting from scratch in that art.