The nature of information often makes it look like a public good. Is this the case for certification of peoples abilities? Is there a role the government could play in certifying people's abilities? Maybe do people thing that that's one of the roles that public universities play? Spence I guess is the typical reference that suggests that people go to school for certification/signaling of abilities rather than to acquire skills. Maybe that's a reason that there are many public universities?
Information is certainly not a public good. Job applicants know their skills with certainty, while employers do not know the skills of applicants. The application process is designed to increase the employers information set so that they can select the best candidate. However, applicants have an incentive to lie, so this process may not always be effective.
Certification, on the other hand, is a credible quality signal that cannot be (easily) faked. This massively reduces the employer's search and opportunity costs. This is not just theory: many job applications simply require a university degree, and virtually all require at least some type of certification.
Whether the government has to play a role in the certification process is another question. An ideological rather than an economical one, if you ask me. Regardless of the answer, it should be noted that many advanced certification standards are upheld by non-governmental bodies. Even if public universities are considered part of government, they are usually accredited by non-governmental organisations. And so are most driving schools, cooking schools, professional degrees, etc.
The question seems to have changed a lot since this answer.
The appropriate role of government is to improve on free activity of the market by improving how the market works (ie fixing market failures). For skill certifications to be an appropriate role for government, such certification would have to be a public good, which would require that it creates a positive externality - a value given to someone that hasn't paid for it.
In the case of skill certification, there certainly is no positive externality. It's easy and in fact very often done in the free market. There are a myriad of certification programs, schools that give degrees, and tests you can take that provide some proof of skill. All of these things can be done with your usual buyer-seller relationship without any externalities happening. Therefore the government doesn't have a role to play there.
I wrote a whole article on the appropriate role of government, which might clarify things for you: https://governology.wordpress.com/2016/07/05/the-role-of-government-part-1/