Yellow Kid is indeed widely considered to be the first "comic character" though his "strips" were typically single-panel gags. He was also one of the first characters to be licensed for product advertising, and you'll see his image in lots more places than just newspapers and buttons of the period. The buttons you saw on eBay look to be from a numbered series of 155 Yellow Kid buttons produced in 1896-98, though I can't make out the numbers in the photos due to the bad discoloration. The buttons are listed in Ted Hake's button books, and you could probably match them up to better-looking exemplars if you dig enough in the Hake's Americana completed auction listings. Most of the other buttons in the series have country flags or generic sayings, so ones referencing a specific university would likely carry a premium.