Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your guess. As I was standing in line the other day I realized that their advertisement of "make it a large to double your chances of winning" (where large drinks/fries have two pieces on them) was actually not exactly true. The real probability of having at least one winning with two tickets is $\frac{5}{9}$ rather than $\frac{2}{3}$ which would really be double the chance. This is because the probability of both losing is $\frac{2}{3}\times\frac{2}{3}$ so the probability of at least one winning is $1-\frac{2}{3}\times\frac{2}{3}=\frac{5}{9}$.

Now this was all very clear to me but while I sipped my strawberry banana smoothie, I wondered why this game of two three-slot tickets where each one has one winner and you can scratch one from each is different from the game of a single six-slot ticket where there are two winning slots and you get two scratches. The games must be different because the probabilities of winning are different. The six-slot game has $1-\frac{4}{6}\times\frac{3}{5}=\frac{3}{5}$ chance of getting at least one win. The two games seem the same to me intuitively. Can anyone explain how they are different?

EDIT: I had noticed that in the two tickets, by making one guess, you actually eliminate 2 other possibilities with it so maybe this is the core of it, but I am still trying to see it more intuitively.