When we lived in CA the school cutoff was Sept 1. We moved to CT the year before my son started school and the cutoff was 12/31 of that year. My daughter had a bunch of late-year-birthday kids (who were 4) in her Kindergarten class. Often parents delay starting school for their sept, nov, dec bday kids, especially if they're boys, to give them time to mature.
I assumed Jr turned 13 at some earlier date (since they didn't show a family party for him) but just learned about bar mitzvahs or just decided then he wanted one. His school maybe a grades 6-12 school, (especially if it's private, divided into a lower and upper school), so he may have become aware of bar/bat mitzvahs from a child in the lower school.