I suppose it is mildly related to the 11th century.
In Polish education, it is strongly underlined that Mieszko I was a Książę (Duke, Prince) and not a Król (King) like his son, Bolesław, who was born into Christianity, unlike his convert father, and who managed to get the crown from the Papacy in the year of his death, in 1025.
So, first of all: Why did he adopt Charlemagne's name as his title? Król is widely believed to come from Karl/Karol (see Russian Король). But Książę/Knyaz' is believed to be a cognate with English King, and to have been an equivalent title, with titles translated as Grand Prince being like the Celtic High King, a king of other kings. Had other Slavic rulers converting to Western Christianity done it before?
Why did Charlemagne's name replace titles like Książę, Князь or the Lithuanian title of Kunigaikštis, even eventually in languages of Orthodox Slavs who seem to have kept these titles after conversion?
And what was exactly the importance of being crowned by the Pope? When did this precedent start? When did it end? Was there a similar practice among Orthodox Christians?
This may be simply a case of linguistic confusion on the part of the Poles. There's precedent elsewhere in the period: the German bishops and chroniclers Adam of Bremen and Thietmar of Merseburg disagree during the eleventh century as to whether the Wendish 'Radigast' is a person, a god or a city, for example. With Charlemagne styled as Karolus Rex or Imperator Augustus, it is not inconceivable that "Karol" was accidentally appropriated as a title, or actually done purposely in order to display an aspirational allegiance to the Holy Roman Empire.
Especially following the Schism of 1054, receiving your crown from Rome was particularly symbolically important. In part it fed into concepts of Romanitas, a cultural notion of legitimacy and authority deriving from the continuation and propagation of Roman Imperial laws, customs, symbols and religion. This was actively encouraged by the Church and the theology of Augustine of Hippo's City of God which postulated Christendom as the spiritual successor to the Roman Empire, with a duty to unite and guide its people. Just as Roman political, military and econonic power was an aspirational goal, so was membership of the Roman Church.
Charlemagne himself had been crowned by the Pope, as more recently had Otto I, II and III, whose resurgent Holy Roman Empire was encroaching ever further eastward. Securing a Papal coronation was a statement of Polish political autonomy and faith, a ward against any Germanic designs on Slavic, 'Pagan' territories. St Stephen of Hungary had similarly received his crown from Pope Sylvester II in 1000, converting in a way that would build alliances with the Holy Roman Empire who had previously been enemies of the Hungarians. The Catholic option is equally symbolic. Accepting the authority of the Lateran was a powerful symbol of autonomy from the Orthodox Church in Constantinople, which was much closer and had designs on extending its authority into Eastern Europe.