Nope, an ecumenopolis is not a religious city, like the Vatican, but rather a ‘world city’, from the Greek which means ‘world city’. The ecumenopolis is the term given to a planet-spanning urban center, and while there are pretty clearly none on Earth (since you would otherwise know about it), ecumenopoli are often featured in science fiction.
But does the idea of an ecumenopolis even make sense? From a metabolic viewpoint, unless there were off-planet raw materials and foodstuffs being created, this kind of city does not seem viable. In general, cities are not self-sufficient, so unless we become a multi-planet species, or drastically modify the nature of the city to include mechanisms for agriculture, mining, and so forth, this will remain an impossibility.
And this is aside from the political aspects. Most likely, even though it will be an urban environment spanning an entire planet, this will not be a unified one, with a single municipal government. An ecumenopolis is just too large for that to work (at least as I can envision it, early-21st century observer that I am).