I've only seen the two movies and little bits and pieces of the animated series and didn't spot any mention of an other genie. So, is Genie actually, THE Genie, one of his kind until fulfilling Jafar's third wish, or is he a part of a family/pantheon? Tied to this are the following: Was Genie born a genie or was he turned into one like Jafar? Who was the original Genie?
Looking through the full list of Aladdin characters, the genies that we see on screen are Genie, Evil Genie, Jafar, and Eden. While we know that Jafar was originally human before wishing to become a genie, there's nothing that I can find that alludes to Genie's or Eden's life before making their appearances. In fact, the very first sentence of Eden's bio starts with:
Not much is known about her background...
Evil Genie is just a shadow copy of Genie, so we can say that his origin story is being cloned through magic. It should be said that Eden is the only one to live in a bottle, while Genie and Jafar have lamps. This could be because her name is likely derived from Barbara Eden, who played a genie in a bottle in the 60's sitcom I Dream of Jeannie.
Genie does play The Peddler that narrates the first movie, and the linked list page has this to say about him:
In the original ending of Aladdin, the Peddler was revealed to be the Genie masquerading as a human, but this was cut from the final film. The directors of the movie later revealed that the Peddler is still the Genie, despite the ending being deleted.
So while appearing to be a separate character, they are actually the Genie.
While we only see four genies throughout the movies and TV shows, we do know that there are more in the universe. The character, Mukhtar
is a reptilian humanoid, apparently the last surviving member of the race of Mukhtars, sworn enemies of Genies.
From this we can safely assume that there have been, and possibly are, many more genies flying around. We only ever see the four though.