Indiana State plays football at the FCS level.
The California system always seemed odd to me. You have an entire collection of schools that are within the University of California system, and then another whole collection of schools in the California State University system. There is little argument that the UC schools are the premier schools, and CSU the second tier, when it comes to admissions, academics, etc. And it's true in football as well, where all 4 FBS P5 programs in the state are either private schools or UC schools (USC, Stanford, UCLA, Cal). There are 37 college football teams in CA, but the only two other FBS programs are Fresno State and San Jose State.
And of 9 schools in the UC system, only two of the 9 field FBS football teams. Two more field FCS teams, and the remaining 5 don't have football teams at all.
Texas is similar. There are many universities within the University of Texas System (The University of Texas at Austin, UT-San Antonio, UT-Dallas, UT-El Paso, etc.). And then the Texas A&M University system has its own collection of schools (Texas A&M in College Starion, Texas A&M Commerce, Texas A&M Corpus Christie, etc.). Only one of the Texas A&M system schools plays D1-A football (the original Texas A&M in College Station), but three in the UT System do-- UT, UTEP, UTSA.
To address Cincy's original question, in a lot of states, the "State" university started as the ag or land grant school, and the "University of" was the mainstream or flagship university. Oklahoma State was originally Oklahoma A&M. Kansas State University was originally Kansas State Agricultural College. There are some other examples. As those colleges expanded beyond just their ag roles, many of them wanted their names to reflect their broader mission, so they changed to "State" rather than A&M. This didn't happen EVERYWHERE, as obviously Texas A&M never made that change.
Also, specifically for Texas State, that school was called Southwest Texas State for a very long time, until the early 2000s. It's an independent school, not from either major university system, and when folks started turning their noses up at "directional" schools, Southwest Texas State decided to rebrand as Texas State. Other directional schools have rebranded similarly, like Southwestern Louisiana that decided to rebrand as the University of Louisiana Lafayette.