No: the linked article requires that further questions split the previous groups each in half. The first question must split the group in half, the second quest

No: the linked article requires that further questions split the previous groups each in half. The first question must split the group in half, the second question must split those two groups in half, and so on. When we hit the last bit, the final question is splitting groups of 2 people into the last bit, and each person has been assigned a unique number. In this manner, 33 bits is enough, and the birthday problem is avoided. The page mentions all this.