No, but there may be many human slaves living in the lamp.
While that line is puzzling, there are no other references in the canonical story to the nature of any additional supernatural slaves (jinn or efreet) in the lamp. However, the genie of the lamp frequently produces human slaves, male and female, black and white, to serve Aladdin, his mother, and the sultan. These may have been the slaves that the genie was referring to.
The earliest version of the story was published by the eighteenth-century French Orientalist, Antoine Galland. He added a number of additional tales to the original 1001 Nights, based on stories and documents that he heard and read in the Middle East. While some of the stories he added have been documented in earlier Arabic versions, Galland's most famous addition to the Arabian Nights canon, the story of Aladdin and the Wonderful Lamp, is not known to exist in any earlier editions than Galland's. (As a result, there is still some debate over whether it is actually an authentic Arabian folktale.)
This version is a direct translation (by Henry Weber) from Galland's French text. The first appearance of the genie of the lamp occurs on page 347, with the mention of additional slaves present. (The genie of the ring had previously appeared and freed the protagonist from the tomb where the lamp was found.) Subsequently, there is no mention of any other genie being called forth from the lamp.
However, there are, as previously mentioned, many human slaves (dozens, at least). There is no mention of where these come from (except that they are brought by the genie), or where they go when they are not actively serving the master of the lamp. Therefore, it is at least plausible that these human slaves are the ones the genie is referring to.
While not generally a faithful adaptation, this anime version of the story of Aladdin includes the humans slaves.