To go a bit further, did there used to exist regional discrepancies in the rules? And do those places still play that way, or has the standard rule set taken over?
The short answer is books, published in the 15th and 16th centuries CE, in Spain and Italy.
Chess spread through trade and war. Starting in India in the 6th century CE as chaturanga, chess was introduced to China, likely by traders or Buddhist monks, where it influenced the development of Xiangqi, but "Chinese chess" has stayed distinct, as has the later Japanese variant shogi. Our current variant of chess comes from India's trade with the Sasanian Empire (Persia), where chess was called chatrang then shatranj. Some of our terminology in chess still owes to this period, such as checkmate (shah mat--the king is helpless). From there, chess would spread directly to modern-day Russia (who still calls chess shakhmaty).
Through the Muslim Conquest of Persia (7th century CE) chess would spread to the Arabian Peninsula and into modern-day Egypt. After a civil war, the Umayyad Caliphate would extend its reach over North Africa and into modern-day Spain, bringing chess with it. We think the Moors introduced chess into Spain around the 10th century, where the name would evolve from shatranj to shaterej to acedrex to the current name ajedrez. In Spain, as I've previously explained, the rules were revised toward the end of the 15th century, notably regarding the minister or vizier becoming the queen, with far more influence.
Simultaneously, chess was introduced to Southern Europe from the Byzantine Empire. The Greeks called it zatrikion while in Latin it was formally called Ludus Scaccorum or informally scacci, which became the Italian scacchi, the French "echecs", etc. So when Spain changed the way the queen moved, Italy dubbed this a chess variant they called alla rabiosa (with the madwoman, or Mad Queen's Chess).
But if you wonder when chess transitioned to become more standardized, the answer is around 1475 CE, in Spain. It did this through a book: Luis Ramirez de Lucena published a book called Repetición de Amores y Arte de Ajedrez con 101 Juegos de Partido in 1497, and some of those 101 games were played under the old rules (vizier) while others were played with the new rules (queen). This book would prove highly influential, and later authors would largely follow the rules laid out in Lucena's book. Examples include Pedro Damiano's book Questo libro e da imparare giocare a scachi et de li partiti, first published in Rome in 1512, then Ruy Lopez de Segura's book Libro de la invención liberal y arte del juego del Axedrez published in Spain in 1561, etc.
The game would continue to evolve, and sometimes holdouts would cling to previous iterations, notably when en passant was created, castling was defined, and pawn promotion was clarified. But since the 20th century, and the creation of the Fédération Internationale des Échecs (FIDE), chess has remained consistent. FIDE acts as chess' governing body, and rule changes made by FIDE instantly apply in international competition.