Until fairly recently, most quilts had a muslin backing.
While some people like everything to match, I'm more of a use it up and move it out kind of person. If the quilt has blue in it and I have a large piece of blue, it'll get used. Mind you, my quilts are scrappy style - so lots of fabrics in each color - but if it's in the same color family, I'll use it. If I don't have enough (most of mine are big bed quilts), I'll piece a backing with several fabrics.
While it's doable, I'd be thinking hard about whether it's really worth the extra work. And money, as you'll have a piece of fabric that won't be used.