This won't be a particularly detailed answer, but the question isn't actually too complicated. While Pope Gregory did commission the Gregorian calendar, it was heavily based on earlier calendars with adjustments to the number of leap years occurring in a century and the removal of a few days to make an initial adjustment.

Ultimately the Gregorian calendar is based on early Roman calendars (and perhaps those are based on the Greek lunar calendar). In these pre-Julian calendars, there were ten months with winter lacking month divisions altogether. Under this system the months were Martius, Aprilis, Maius, Iunius, Quintilis, Sextilis, September, October, November, December (Michels, The Calendar of the Roman Republic). You can see an obvious pattern with the months after June (Quint, Sex, etc.). To be honest, I don’t know why the pattern randomly starts at the fifth month, maybe all of the interesting stuff happened in June and earlier. Later, the months of January (named after the Roman word for door, as in door to the year) and February (named for the process of purification) were added to the calendar to extend it to a full lunar year. The ancient sources seem divided on when exactly this happened, some attributed it to the successors of Romulus (who may be entirely mythical), and others to 5th Century BCE decemvirs (Jörg Rüpke, The Roman Calendar from Numa to Constantine: Time, History, and the Fasti ). In any case, the later months kept their names, despite now being named incorrectly. Notably though, March was still considered the first month of the year (despite January’s name) in some capacities for quite some time. For example, officials took office on March 1 until 153 BCE when they began to assume office on New Year’s Day.

Later reforms largely preserved the names of the months as they had been in pre-Julian calendar. The Julian calendar added days to bring the total from 354 to 365 (so that the lunar calendar approached a solar calendar) but retained the names of the lunar-based months. Quintilis and Sextilis were later named in honour of Julius and Augustus. Eventually, due to the drift of Easter through the year, Pope Gregory XIII commissioned a new calendar to iron out this wrinkle in the system. However, Papal authority did not extend to changing the entire world’s calendar (as evidenced by its slow uptake throughout Europe). Pope Gregory was entirely dependent on civil authorities making the switch themselves (Meeus and Savoie, The history of the tropical year). For this reason, and to avoid complications, the calendar’s divisions largely remained the same with adjustments being made to the number of leap years.