Answer:
No, it is not. It is a German word that means "the spirit of the times", in the sense of the general intellectual or moral characteristics of a given time in history, including the culture and tastes of people living at that time. It is used in English, and is usually not capitalized.
However, when you capitalize the word, you are actually talking about The Zeitgeist Movement (www.thezeitgeistmovement.com). This isn't a religion because mostly, it doesn't believe in anything. It's mainly a social group of activist who have belief on how we could make the world a better place.