What is the absolute first intercollegiate sporting event that occurred?

The absolute first intercollegiate sporting event probably can not be determined, as it would have been any unsanctioned contest between attendees of different schools. A simple footrace or wrestling challenge would qualify.

The first organized intercollegiate event in the U.K. was the first University Match at cricket between Cambridge and Oxford in 1827, pre-dating the premiere of their more famous Boat Race by 2 years:
https://en.wikipedia.org/wiki/The_University_Match_(cricket)

The first organized intercollegiate sporting event
in the U.S. was in 1852, a rowing race between Harvard and Yale. https://en.wikipedia.org/wiki/College_athletics#Beginnings

The first American intercollegiate sporting event took place on August 3rd, 1852 when Harvard and Yale competed in a two mile rowing competition.

On August 3, 1852, three crews from Harvard and Yale met on New Hampshire's Lake Winnipesaukee to race against each other for the first time. In this two-mile (3 km) contest, Harvard's Oneida prevailed over Yale's Shawmut by about two lengths, with Yale's Undine finishing third. The prize for first place was a pair of black walnut, silver inscribed trophy oars. General Franklin Pierce, who would later become the 14th President of the United States, was in attendance. Since that day in 1852, the Yale-Harvard Regatta has been contested an additional 146 times in various locations in the northeast, becoming an annual event in 1864.

Sports Illustrated has called this the most venerable rivalry in all of college sports, and as the oldest collegiate athletic event in the country, the Yale-Harvard Regatta is indeed the cornerstone of intercollegiate athletic competition in the United States.

This competition was the first test for the Yale University Boat Club, which was established on May 24, 1843, when a group of Yale students purchased a small rowing shell called the Whitehall. Harvard would found its boat club the following year. The second matchup between the two schools was three years later in July 1855 on the waters of the Connecticut River in Springfield, Mass.

https://en.wikipedia.org/wiki/Harvard%E2%80%93Yale_Regatta