Generalization of Birthday Problem over Sets of Dates

Question

At the classic version of the birthday problem we check for collisions of scalars $ b_i $ sampled from a uniform distribution. How can this be generalised over sets with more than one samples per person? E.g. in the case of two dates (birthday and name day), we will have to pick a set of two scalars $\mathbf{b}_i = \{b^1_i, b^2_i\}$ for each person. If a collision means that the two sets over two random people are identical, which is the probability of a collision for a population of $m$ people?