What's wrong with the argument that the probability of rolling, a double 6 in two rolls of a die is 1/3 because 1/6 + 1/6= 1/3?

You don't "add" the probabilities, you multiply them.

If the above argument were valid, then a roll of 6 six times in a row would be
1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1
or "absolute certainty", which of course is nonsense.

To show that the above answer is (1/6)(1/6) = 1/36
we could list all the possible outcomes.
There would be 36 such outcomes, and only one would be (6,6)