The only correct answer to this question is 2012-09-10, the PostgreSQL 9.2 release date. In that release they unveiled JSON datatypes, and developers were able to have their cake and eat it too.
It's funny because I was going to say circa 2012. Working at Linode I definitely saw a shift away from MySQL to Postgres around that time, but I didn't have any over arching reason beyond "it was trendy."
I think CTEs and window functions in version 8 were big, combined with udf you could write some very complicated queries for data analysis without exponentially growing number of joins.
https://www.postgresql.org/about/news/postgresql-92-released...
https://wiki.postgresql.org/wiki/What%27s_new_in_PostgreSQL_...