I've heard some European folks refer to certain shooters as "Ego Shooters". Primarily in reference to first-person shooters, but I've since heard it used in reference to non-FPS titles as well.
Ego-Shooter is just the German translation for First Person Shooter. There may also be other ways to say it, but this one seems fairly common.
Most things don't translate perfectly from language to language if you go a single word at a time.
An Ego-Shooter is a First-Person Shooter. It is used in countries where the literal translation would be cumbersome or not conveying the meaning properly.
Ego as a prefix stems from Latin respectively Greek and means "I" or "Self". It is used to convey the specific meaning; that you are the avatar and control it directly.
To highlight this further, other languages have different terms: in French for example, it is called "Jeu de tir en vue subjective" (Shooter played from a subjective viewpoint). However, many languages stick with the literal translation from English (Italian, Spanish, Portuguese), while others simply stick with the English term (Dutch, Danish). the abbreviation (FPS) is also used across languages as it is easier to use than the proper English term for many non-English speakers.
As for it being used for non-FPS games, the term is mainly used to categorise a genre of games. Genres are inherently subjective and since the term resides in the domain of popular culture, its application may sometimes diverge from the original meaning. Unfortunately there is no way I could further comment without getting some concrete examples.