I think there was too many charlatans in the past and still too many today. But these guys are for real and you need to show this quality.
Also, the Ghostbusters received many negative publicity and doubt back then; I think the city would want to make it clear that these guys are to be taken seriously… considering the fact that some people today still believe that Earth is flat, that we never walked on the moon, and that no planes crashed on september 11. I'm sure the statue of liberty changed the general opinion in gb2, but those with doubts can still influence the public; it's a ghostbuster company after all. This badge would be a good move from the city to authentify the ghostbuster's professionalism.
As a company dealing with dangerous unknown phenomenons, and recognized by the city, you need to be able to close instantly the zone you want to the public when needed. For that you need to have some authority and this badge represents also the city behind you.
Ghostbusters is a “cleaning service” but it's far from dealing with “trash”. People who deal with paranormal activity cannot be considered as “regular men” with an ordinary “cleaning job”. They can be almost considered a mix of police department and firefighter; they extinguish the “fire” in one place, capture its factor and imprison it after; it's a dangerous job, a level under or above the police/fire department, which gives to the metalic badge all its meaning.
If it's not enough then I'll say that Peter Venkman would demand recognition for the company's professionalism and support over the years, asking to be officialy a part of the city's security services. If he's not receiving it, I guess he would order those shiny badges to not only congratulate the efforts of its team, but to heal the wound made by the fact that saving life doesn't make you a superstar or someone more respected. This badge would represent all the hard work made.