We don't usually see "Stands" with the same ability. How does Jotaro develop a power similar to Dio's in the final battle. And now, everytime he uses his stand, he shouts out "Star platinum the World" as if he adopted Dio's power. What happened in the final fight with Dio?
While Star Platinum has a bad habit of pulling new abilities from underneath his loin cloth (I'm looking at you, Star Finger), I don't think it's the case here. I think it was implied that Star Platinum had the Time Stop ability the whole time.
When Jotaro confronted Dio, he quickly figured out that their stands were very similar in terms of speed, range, strength, size and what not. And once Kakyoin figured The World's secret ability, it dawned on Jotaro that Star Platinum might have a similar skill.
Of course, he couldn't trigger it easily. At first he was merely able to see glimpses of the stopped time, then he's able to move in it for a moment and only at the very end does he finaly manage to stop time on his own.
Not all stand abilities are known to their users instinctually. Even Dio discovered the Time Stop by accident while training. He also had to train his ability to extend its length and his movement in it.
While never actually explained, the most likely reason their stands are the same type is their macabre family connection. Jotaro shares a bloodline with the body Dio currenlty posesses (through the world's first vampire to human head transplant).
There are two explanations. A in universe explanation, and a meta explanation.
The in universe explanation, is that when Jotaro realized The Wolrd's ability of time stop, he thought that maybe since his stand's ability is super-speed, the next logical upgrade from super-speed is time stop, and so he tried it out, and it worked. I'd like to add that it is possible that Jotaro inherited the stand from Dio, through Jonathan's body, just as Joseph inherited Hermit purple, from Jonathan. However this is speculation, and no-where is it confirmed. Furthermore, some fans argue that Time-Stop had been foreshadowed throughout part 3, especially in situations like the Darby fight, where he demonstrates Star Platinum's insane speed. Some theorize that Darby knew The World's ability, and so he was so panicked by this demonstration, he thought Star Platinum is the same. Once again, I clarify that these are only speculations.
The meta explanation is that Araki had thought of The World as the very first stand to be conceptualized, followed by Star Platinum1. This could mean that he intended Star Platinum to have the ability from the start, to fit the the 'fight fire with fire' trope.