DVDs have an upper layer of plastic (poly substrate). It's impossible.
Whoever told you ink "bleeds through" does not know any better. If that were true, the pen would be eaten too, and could never be sold. These are markers, not vials of hydrochloric acid.
That is an old myth, it started in the days of CD-R. Now, CD-R did not have an upper plastic layer, and could be damaged more easily, especially the unbranded ones that lacked a frosted coating. But it was still a myth then.
There is no correlation between ink from Sharpie markers, or any other markers, harming discs. Just another boogey man.