Usually, an issuing authority that runs the certification program obtains a trademark/service mark over its designation (e.g. CFP(R) for Certified Financial Planner), and then licenses its use exclusively to people who have earned the designation, administers eligibility for the certificate, and polices misuse of the trademark/service mark via cease and desist letters and civil lawsuits. This has also been done by a professional association for the professional title "Realtor(R)".
Similar approaches are used for vendor certifications such as UL for Underwriter's Labs and BBB for Better Business Bureau approved.
In the absence of other legislation governing professional regulation in the field, no government approval is necessary to set up the scheme. But, there are some regulated professions (e.g. law and medicine) which routinely regulate the advertising of specialities and certifications by members of their professions who are regulated.
None of these would be proper to use on an official identification card in most cases.
This is much less strict than, for example, Germany, where this kind of thing is tightly regulated.