In the US MSC has a different approach to cruising than in Europe. In the US the cruises have a fixed starting and end port, while in Europe they don't. In Europe it is more a continuous service. Meaning that in Europe in almost every port some passengers disembark and new embark. This makes sense in Europe, because for example a Spanish person would not want to travel to Genua to embark, for a cruise that calls also in Barcelona. Therefore in Europe a back to back is most often a continuous cruise (although Genua is somewhat of a focus port, with a sailawayparty and a goodbye show the day before). In the US you're probably be required to leave and board again between cruises, even if sold as one 14-day cruise.
As for testing, some governments (like the Italian) require all passengers 6+ who are not vaccinated to be tested before entering from a non-EU country. Since all passengers 12+ must be vaccinated to board anyway, this only applies for 6-12 year olds. The testing is done in the ships medical centre, is free, and rapid.
