The question is not about there being a limit to people who could wish stuff, or even things to be wished for.
It's that Aladdin promised Genie that he'd use his last wish to set him free.
ALADDIN: Fair deal. So, three wishes. I want them to be good. (To GENIE) What would you wish for? ...
ALADDIN: You're a prisoner?
GENIE: It's all part-and-parcel, the whole genie gig. Phenomenal cosmic powers! Itty bitty living space.
ALADDIN: Genie, that's terrible.
GENIE: But, oh--to be free. Not have to go "Poof! What do you need? Poof! What do you need? Poof! What do you need?" To be my own master, such a thing would be greater than all the magic and all the treasures in all the world! But what am I talking about, here? Let's get real here. It's not gonna happen. Genie, wake up and smell the hummus.
ALADDIN: Why not?
GENIE: The only way I get outta this is if my master wishes me out. So you can guess how often that's happened.
ALADDIN: I'll do it. I'll set you free.
GENIE: Uh huh, right. Whoop!
ALADDIN: No, really, I promise. After make my first two wishes, I'll use my third wish to set you free. (He holds out his hand)
GENIE: Well, here's hopin'. (Shakes ALADDIN's hand.)
By the end of the movie, Genie and Aladdin have become friends. Friends enough that when its time for Aladdin to make his third wish, and it looks like Aladdin and Jasmine may not actually get to be with each other because of the law, Genie offers to forgo his freedom for the two of them.
ALADDIN: Jasmine, I'm sorry I lied to you about being a prince.
JASMINE: I know why you did.
ALADDIN: Well, I guess...this... is goodbye?
(GENIE pokes his head around the corner shocked at what he is hearing.)
JASMINE: Oh, that stupid law. This isn't fair--I love you.
GENIE: (Wipes away a tear) Al, no problem. You've still got one wish left. Just say the word and you're a prince again.
ALADDIN: But Genie, what about your freedom?
GENIE: Hey, it's only an eternity of servitude. This is love. (He leans down next to her.) Al, you're not gonna find another girl like her in a million years. Believe me, I know. I've looked.
Sure, Aladdin could have wished for something else and passed the lamp around. And hey, Genie's waiting 10,000 years, what's 5 minutes more?
But it would prove to Genie (and the audience) that Aladdin
Has no respect for Genie as a person.
Sure, it's terrible that Genie has to serve others and can't be his own master, but he can be his own master once he's finished serving all my friends, that seems cool.
Gives no hoots about fulfilling his promises.
His promise was to use his third wish to free Genie' not 'I'll force some other shmuck to give up their third wish to free Genie'
But then, what assurance does Genie or Aladdin or anyone have that whoever ended up being the last in the Genie-wish train would use the third wish to free Genie? Genie's best rapport in the group is with Aladdin; if even that guy wouldn't give up his third wish to free his friend, why would anyone else? Everyone would always have that just one more friend for Genie to fulfill wishes for.
So whatever kind of person Aladdin would have to be to turn around and do this, that is not the kind of person we saw Aladdin being in the movie, and Aladdin was not willing to pretend to be someone he was not, any longer.
And, tbh, that's not the kind of person movies would ever uphold as a Heroic character (which Disney's Aladdin is supposed to be) either.