I've heard it rumored that Tigers were actually a group of southern sympathizers which of course something any modern brand would want to distance themselves from or perhaps even rewrite in the case of MU, if true.
Also a fun, but unsubstantiated note on the name Jayhawks: Lawrence is atop the highest hill in Kansas so they always had a birdseye view of anyone coming, like a hawk. They'd then flank them, in the shape of a J. For this reason Quantrill raided at night under the cover of darkness and massacred anyone in town.
Both universities acknowledge their nicknames derive from Civil War militias. Kansas proudly claims the Jayhawkers were freestaters while Missouri declines to mention which side the Missouri Tigers fought for. But if you read up on Jennison's Jayhawkers they aren't exactly paragons of abolitionist morality and drew their fair share of criticism from Union supporters. Also, Missouri's Tigers were probably the militia in Columbia who blockaded the city to protect it from the infamous Confederate Bushwhacker "Bloody Bill" Anderson rather than free-stateist Jayhawkers.
Anyways, the other Tigers:
LSU, is in fact, Civil War related, according to their first coach:
"It was the custom at that time, for some occult reason, to call football teams by the names of vicious animals; the Yale Bulldogs and the Princeton Tigers, for example. This is still the vogue. It struck me that purple and gold looked Tigerish enough and I suggested that we choose “Louisiana Tigers,” all in conference with the boys. The Louisiana Tigers had represented the state in Civil War and had been known for their hard fighting."
Auburn, the other Tigers of the SEC, took their nickname from the line "where crouching Tigers await their hapless prey" in The Deserted Village a 1770 Oliver Goldsmith poem. However, college sports fan will be familiar with Auburn's incongruous mantra/cheer "War Eagle". The phrase has multiple origin stories but the most popular, if improbable, origin has a Civil Waw connection. The story goes that a Civil War veteran found a live eagle on a battlefield and kept it as a pet for the better part of 30 years. When Auburn played Georgia for the first time in 1892, the veteran was in the stands and the eagle broke free and soared around the field as fans chanted "War Eagle". At the conclusion of the game, an Auburn win, the eagle fell from the sky and died on the field.
The other two southern FBS Tigers have pretty boring origins, Clemson's is an allusion to Princeton and Memphis derives from a cheer.
The remaining 4 Division 1 Tigers in the former Confederacy (Grambling State, Jackson State, Savannah State and Texas Southern) are all HBCU's and though I couldn't find much information on the history of their nicknames, I think it's safe to assume they aren't named after CSA units.