The practice of a king having his son crowned during his lifetime was not uncommon; perhaps the most famous example is Henry the Young King, son of Henry II of England. This was usually done in an attempt to secure the throne for the monarch's chosen heir by reducing the chances of a contested succession.
In the case of Sigismund II Augustus, his father Sigismund I the Old and his formidable Queen Bona Sforza had a particularly strong reason for having their son crowned early. Poland had a long tradition of the nobility electing the next king and, although they usually followed the hereditary principle, one of Sigismund I’s predecessors, Władysław III, had been elected to the throne against much opposition, despite being the eldest son of his predecessor.
This power to elect the king gave the nobility considerable leverage over the monarchy. According to the Wiki articles on the Jagiellonian dynasty and Bona Sforza, this power of the nobles was something which the Queen, in particular, saw as a threat to "her personal and dynastic interests". According to Jerzy J. Lersky in Historical Dictionary of Poland, Bona Sforza
had a great influence on the King and worked at strengthening the
Polish royal power. She arranged the Royal election of their son,
Zygmunt August, during the reign of his father and built a powerful
The case of Sigismund II Augustus in 1530 was controversial as his parents had to persuade the nobles to break the rule of not electing a successor during a monarch’s lifetime (thus brining in Vivente rege). When Sigismund II died in 1572 without leaving an heir, the nobles adopted the Henrician Articles which, among other things, stated that the elected King’s children “had no right of inheritance with regard to the throne.”
Note: On education, the issue concerning 'proper education' which you mention in the question is indeed unconvincing. Rather, it was the nobles opposing the Queen who were blaming her for the bad education of her son. Thus, education seems not to be relevant here.