For example, in my home state, California, the University of California system is home to two of the most prestigious universities in the world (in academics and athletics) and several others with top rankings, and is generally considered much more selective than the equivalent California State University system, which, with the exception of a few more well-known schools like Cal State Fullerton, tends to cater to local students, and is generally less prestigious. Many other states also feature this system.
EDIT: As a couple of commenters have questioned how common this system is. It exists in more or less this form in the following states
So we can separate this into two distinct questions really, one with a much more straight forward answer, the other less so.
The first is the origin of the trend for their to usually 2 'flagship' schools in each state with distinct naming conventions and differing academic focus in many instances. California is actually really one of the few states to make it extra messy, and it tends to get more messy the further West you get, partly simply because higher education often got a later start there.
That answer tends to be the result of the Morrill Land Grant Acts. First passed in 1862 the scheme provided for the sale of federal land in each state or the territories and the funds used to set up a new school in each state. The schools were to focus more on supporting the working farmer and labourer classes, teaching agriculture sciences, mechanics and engineering, and with a dash of the liberal arts and a general education. Unsurprising for a law passed at the height of the Civil War, the students were also required to get instruction in the military sciences and tactics, to provide at least partly trained officers in times of need(this requirement was one of the fore-bearers of modern ROTC). Many of these schools originally had names like STATE Agricultural and Mechanical School, STATE Technical College, or similar which have evolved with their expansions of curriculum and focus, resulting in names like Oklahoma State University (used to be Ok A&M), or from Agricultural College of the State of Michigan to Michigan State, and some still keeping it old school like Texas A&M. Others took their names from locations, Auburn for instance was at first the Agricultural and Mechanical College of Alabama.
The new schools were meant to be distinct from the existing higher education options in most states, which were often expensive, elite institutions with a focus on liberal or philosophical educations and credentialed professions like the law or medicine. The states also had the option to either eventually integrate the school, or fund a separate school for black students from money in later supplementary Land Grant Acts.
We can take a state which makes the trends I think very clear, Virginia. You have the old school in the University of Virginia, the main land grant school in Virginia Tech, and the HBCU Land Grant Virginia State in Petersburg.
A separate but related issue is the slow death of the Normal or Teacher's college, these institutions, many of which were restricted or focused on women, often had to expand, die, or be absorbed, with the mass influx of post WW2 GI Bill students being a major turning point. So we go from Florida State College for Women, to Florida State University as an example of adapting to survive for a school.
As to why individual states have elected one large, several parallel, or really no overarching structure for their public schools is honestly an answer more about the unique political, budgetary, and social evolution of the government of each state.
My own alma mater, Washington State University, the landgrant school, was founded in 1890, The Univeristy Of Washington in 1861. WSU is the ag school, including the school of veterinary medicine. UW has the medical school (though WSU is getting on now as well). WSU has the extension service, with county extension agents, the master gardener program, and other kinds of research.
The UW is the largest school in the state and is located in Seattle, the state’s largest metropolitan center. WSU is smaller, located in a small college town in the middle of the eastern Washington wheat country.
We also have several “regional” universities that were formerly the “normal” schools: Western, Central and Eastern Washington Universities.
Quick note: I don't believe the nomenclature for Ohio is reversed.
Ohio University was chartered in 1787 by the First Continental Congress, and was established in 1804.
Ohio State University was established in 1870 via the Morril Land Grant Act, and was originally named Ohio Agricultural and Mechanical College.
Vermont seems to conform to this as well
They have University of Vermont (UVM) and Vermont State Colleges comprising a number of campuses across the state, and serving local communities