When and where did Judaism become a religion of the book? Why do the various historic Christian churches treasure various collections of Jewish writings?
A canon or collection of authoritative religious texts may have begun to be defined early, in the 6th century BCE, among Judahite refugees in Babylon. Be that as it may, by the 2nd century BCE, the development of a canon of authoritative texts was in full swing among those who saw themselves as heirs to the promises found in the Torah, the Prophets, and David.
In the first centuries of the current era, across the lands and languages of the countries in which first Jews and then Jews and Christians resided, the inner core but not the outer edges of a canonical standard to which all might appeal was widely acknowledged. This meant that when Jews of diverse provenance became Christians and brought with them the collection of texts their community of origin treasured. some collections were more extensive than others. To this day, therefore, among Christians an Old Testament canon shared by all does not exist.
For a brief introduction to the canon of the Hebrew Bible and three historic canons of the Old Testament, go here.