Do I need to renew my education visa annually if enrolled in a 5-year university program in Thailand?

The UK once had lots of 'Real Universities'. They were mostly just an office running the visa scam until they closed them down and deported the 'students'.

Most real universities give their students documentation to either leave the country and get a 90 day single entry Non-Immigrant Type ED visa from a thai consulate in a nearby country OR they provide documentation for the student to go to the immigration office and apply for a 90 day Non-ED visa inside the country.

After that you get more documentation from the school and go to the immigration office and apply for a yearly extension for 1900baht based on being enrolled in a degree program. You do this every year you're studying. I know people who are on their masters here and have been on yearly extensions since they started the program 5+ years ago.

Is it a Thai (public) government university or a private university?

The documents needed are extremely minimal for a government university, making the extensions a piece of cake!

No, the University will provide documents and help you to apply for an Extension of Stay at local Immigration Office.