What if the twins are monozygotic rather than fraternal? Then the person has two sets of DNA - his and his twin's - but those two sets are identical!

Wow, I love this question, it's an awesome guess indeed.

From studies, identical twins has same sets of DNA, but some environment and conditions can affect the way they're presented in reality.

So for your question, even if it's possible to have a chimera from monozygotic twins, the change/difference will definitely not be significant