(More trivially, there are quines, programs that know full own source code, as you likely know.)

I don't think being able to quine yourself really has anything to do with fully understanding yourself. I could get a complete printout of the exact state of every neuron in my brain, that wouldn't give me full understanding of myself. To do something useful with the data, I'd need to perform an analysis of it at a higher level of abstraction. A quine provides the raw source code that can be analyzed, but it does no analysis by itself.

"The effects of untried mutations on fifteen million interacting shapers could rarely be predicted in advance; in most cases, the only reliable method would have been to perform every computation that the altered seed itself would have performed... which was no different from going ahead and growing the seed, creating the mind, predicting nothing." (Greg Egan).

In any case, if we are talking about the brains/minds of baseline unmodified humans, as we should be in an introductory article aimed at folks outside the LW community, then XiXiDu's point is definitely valid. Ordinary humans can't quine themselves, even if a quine could be construed as "understanding".