The practice of a king having his son crowned during his lifetime was not uncommon; two well-known examples are Henry the Young King, son of Henry II of England, and Philip II Augustus, son of Louis VII of France. 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 (reigned 1504 to 1548) 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
Royal party.
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.”
The declaration as king (usually junior) of the heir apparent has occurred at various times in history. One example is Ptolemy II Philadelphus (died 246 BC), son of Ptolemy I Soter of Egypt. The practice was common among the early Capetian kings of France:
They had their heirs crowned during their lifetime, so that there
would not be even a momentary interregnum. Thus Louis VII, whose reign
is usually dated in the history books to 1137 - 80, had actually been
anointed in Rheims cathedral in 1131, during the lifetime of his
father, Louis VI, and Louis VII’s own son, Philip Augustus, was
crowned in 1179, the year before becoming sole king on the death of
his father.
Source: Robert Bartlett, 'England under the Norman and Angevin kings, 1075-1225' (2000)
Other examples of Capetians crowned during the reigns of their fathers include Robert II, Henry I and Philip I. Interestingly, Louis VI had not been similarly crowned by his father and subsequently found his accession briefly contested by a half-brother.
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.
