Does the string of prayer beads—used for glorifying God or for the remembrance of God, having a specific number of beads (33 or 101) and two distinct beads at which nothing is recited—have a legal basis in Islamic law?

It is permissible to use prayer beads (masbahah) for a need, though it is better to count tasbeeh on one's fingers. However, specifying a certain number for the prayer beads, or adopting a particular method of tasbeeh with them, or remaining silent at their distinctive beads, is an innovation (bid'ah). Every innovation in religion is rejected upon its practitioner.