Similar situation for some years in the Taiwan Strait, where the not-a-republic People's Republic of China faced down the also-not-a-republic Republic of China (Taiwan).
Still, if it's a case of wishful thinking, why "democratic" and not "The Awesome, Glorious Empire of North Korea Death to America"?
So (for a not-quite-correct example) things would progress Congo -> Republic of Congo -> Democratic Republic of Congo -> People's Democratic Republic of Congo.
A "people's democracy" is a country in transition from bourgeois democracy to socialism. In this stage there is more than one class, the largest being the proletariat, led by the Communist Party, which is therefore the dominant power.
The other common country name includes "people's republic." In both cases, it's important to understand the definition of "people". People are communists. Look up "People's dictatorship" for more information on this. Essentially, a "People's republic" is a country that is ruled by a communist party.