HC THREAD!: So basically... you know how the world and star platinum have the same time stop ability? Jotaro says its because they have the "same power type" stand, but he's just bullshitting dio like joseph did kars. In reality, this is why they both have time stopping powers :

Dio, having existed for an eternity and now stuck in a coffin for a century, had a lot of time to think. But even as bright and intelligent as he was, he couldn't stop thinking about how long he had spent and will be spending in his personal prison cell,

the coffin that layed underwater. The crueling torture of being in this coffin must've probably made him not only hate water, but also the concept of time itself. Thus, his deepest desire was none other but to control this very aspect of the universe.

Thus, I think Dio gained the ability of his stand by the wish of controlling the very thing that was driving him insane. As for Jotaro, I do not think he had his time stopping abilities from the start.

I think he unlocked them as the time window of saving his own mother got smaller and smaller, pressuring him into a state of mental instability. He wished for nothing more but to have more time. That is how Jotaro evolved his time stop.

I think them both having time stop is a pure coincidence, as many other close range power type stands exists that clearly do not possess this ability. Jotaro had only realized he had this ability after seeing Dio performing it,

and thus adopted "The World" part as to further piss Dio off, making it seem like he simply copied his ability. Additionally, Jotaro kept it that way after the Dio fight as a memory of a great victory.

END OF THREAD.