In economic terms, government provision of higher education is definitely not a public good. You might think it is a public good because it is good for the public as a whole when people get higher education. However, this is not the definition of a public good.
A public good has to meet two criteria. First, it has to be “nonexcludable.” That is, the person who is providing it cannot prevent people from using the good. Clearly, this is not the case. The government regularly keeps many people out of specific institutions of higher education. You have to apply to the school and be admitted in order to go. You also have to be able to pay. It is very possible to exclude people from consuming higher education. Second, it has to be characterized by “nonrivalrous consumption.” That means that one person’s consumption of the good does not decrease anyone else’s ability to consume it. This, too, does not apply to higher education. There are a finite number of spots available in any given college. If one person enters a college, that is one less spot available for others. Because there are not enough spots for everyone, colleges reject some applicants. This is not nonrivalrous consumption. Thus, government provision of higher education does not meet either of the criteria for being a public good. This means it clearly is not a public good.
What higher education is, in economic terms, is something that has positive externalities. These are benefits to people who are not directly involved in a transaction. When a state provides higher education, it is not just the students who benefit. Many other people benefit because people who graduate go on to get good jobs, pay taxes, start businesses, and many other things that help society as a whole. This sounds like it should be called a public good, but that is not the right economic term.