Can you relabel two 6-sided dices, so that every face has a positive number of dots, and also their sum has the same probability distribution?

Yes, you can do this. The easiest way is to use generating functions. Using simple polynomial algebra, you can see that

(x + x_{2} + x_{3} + x_{4} + x_{5} + x_{6})^{2} = (x + 2x_{2} + 2x_{3} + x_{4})(x + x_{3} + x_{4} + x_{5} + x_{6} + x_{8}).

Therefore, if you take a dice with spots {1, 2, 2, 3, 3, 4}, and a dice with spots {1, 3, 4, 5, 6, 8}, their sum will have the same probability distribution.