Back when the Georgian calendar was introduced (1582) why were the prefixes "sept-" and "oct-" used for the 9th and 10th months rather than the 7th and 8th respectively?

The primary reason is because this is what those months were called in 1581 under the Julian calendar.

The two calendar systems, Gregorian and Julian were not much different. The new system reduced the number leap days by 3 days every 400 years and took a one-time adjustment of ten days. October 4 1582 in the old style was followed by October 15 in the new style.

Protestant dominated countries did not immediately shift. England waited until 1752 to make the change and by then there had been one more leap day (Feb 29 1700), and they made the adjustment in September 1752. If you use a unix/linux computer you can type "cal 1752" to see the calendar jump, unix having its origins in the former colony of New Jersey.

But to get back to the original question, it all goes back to the Roman calendar(s). The original Roman calendar had four months: Martius, Aprilis, Maius, and Iunius, which of course correspond to March, April, May and June. The start of the year for religious purposes was March 1. The rest of the months were just numbered, Quintilis for the fifth month through December for the tenth.

The length of these months was left to the wisdom of the priests (pontiffs) who sometimes added or subtracted days, and they also decided how to measure the remaining time between the end of December and the Kalends of March. Eventually they got into the habit of assigning two months Ianuarius and Februarius plus sometimes a few days that were not part of a particular month or an entire extra month. Every year was different, and was often subject to political influence. In 46 BC the Ponitfex Maximus was Julius Ceasar (yes, the former general and future corpse) and he announced a new system of a standardized calendar. It had some adjustments to make sure that the festivals landed in the right part of the year. (It also had the effect of making his annual tenure as high priest last for 445 days.)

Rome was governed by an annually elected pair of Consuls, and the years were not numbered so much as named after who was consul at the time. Sometimes the term started in March, sometimes in May, and in the second century BC it got switched to January and settled there. Several centuries later when the Christian liturgical calendar began counting Christmas as the start of the year it reinforced the switch to January.