The chance to have 2 copies of a specific card from your deck after 13 draws is definitely different than the chance of having 2 copies of any card after 13 draws.
First, we need to work out in how many orders you can draw 13 cards to include those 2. So the first could be the first card drawn, or the second, … or the last. And the second could be any except the one the first was. That means that there are 13 x 12 = 156 possible ways to draw 13 cards and have those specific two included. (If we only look at “the card” vs “other card” and don’t care about what other card that is). Except that the two cards are identical so we counted double (since “card found on draws 7 and 12” or “on 12 and 7” are the same). So there’s actually 78 options.
For each of those 78 ways, the chance that it happens is computed effectively the same. E.g. to get those two as 12 and 13, the chance is 28/30 (since the first draw has 28 “good” cards out of 30) * 27/29 * … * 18/20 (for the 11th card) * 2/19 * 1/18 (for the final two cards that are the ones we want). This can be rewritten as:
(28 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 2 * 1) / (30 * 29 * 28 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18). For each other combination, the order of the terms in the multiplication changes, but the terms are the same; since order is irrelevant in multiplication that means that the above formula is the chance for each of the 78 possible ways.
Using rules of simplification, we can remove the terms that are on both sides of the division, for a much simpler actual calculation: (2 * 1) / (30 * 29).
And since there are 78 ways, each with a (2 / 870) chance of happening, the chance that one of them happens is 78 * (2 / 870) = exactly 20%.
If you put two copies of a card in your deck and you need to have both in your hands after drawing 13 cards, then that will happen in one out of five of your games.
Obviously, smart mulligan and running card draw in the rest of your deck will significantly improve your chance. And adding survivability cards will help you live longer then 13 card draws, which will also increase the chance to get those specific two.