While we don't really know how many biological processes work, we have a pretty good idea how the basic mechanics of life work. The discovery of "something akin to DNA fingerprinting that lets people discover ancestry" would be the type of major discovery that would completely upset our understanding of things. Such a discovery would have to be consistent with all the evidence we already have (and is explained by our current understanding) while simultaneously introducing entire new concepts.

Not strictly impossible of course, but very few things are impossible in the strictest sense. For all practical purposes, given our understanding of the world, "there is no imaginable technology that could do this" is correct.