Why can't Fiat be issued by random people?
It's of course easier for government to force people to accept their fiat, but that is a technical issue. There are other ways to make people accept Fiat. Religious belief, popularity of the issuer, fear of exclusion (which is kind of force).

And what is a government? If a town creates it's own currency, is it not fiat? What about if the GNOME foundation votes on a new currency?

Fiat is Money by decree. It's easier for government to create Fiat. But it's not the only way