A Gemini is someone who was born while Sun was moving through Gemini: late May to late June, every year Regardless of the year, regardless of the time of birth, regardless of the place of birth
the Sun Sign is only 1 out of 40-50 factors in each person's unique birthchart, and no two Sun-in-Gemini people will be the same.
In fact, what YOU mostly see in a particular Sun-in-Gemini person might have NOTHING to do with their Sun Sign. Often the Sun is NOT the predominant influence on the personality.
Btw, it is not signs but planets that indicate a TENDENCY towards a Narcissistic Personality Disorder.
Sun strong AND badly-afflicted by discordant "aspects" (distances) with other planets CAN be narcissistic.
Also, any of the personal planets (and the more the merrier) in Leo - IF afflicted by discordant aspects - can be narcissistic.
As can a badly-afflicted Saturn in the natal chart.
Or Leo Rising with Sun badly afflicted, or personal planets in the First House badly afflicted.
Astrology is WAY more complicated than just their Sun sign.
Sun signs are easy to find, so people who don't care if they mislead you can make money by writing about Sun signs.
But that isn't astrology.
And if you follow Sun-sign astrology, rather than real astrology based on individual birthcharts, you will mislead yourself and start thinking in stereotypes rather than gaining accurate insight into others.