I surely think it is the way Obito got his MS, but not Kakashi. First time we see him use it is here:
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It is in the fight against Deidara and how Deidara loses his arm. Kakashi says it's a new sharingan, meaning he didn't have it before the time skip. It is a possiblitiy that Kakashi had someone we don't know about that died during the time skip, maybe a girlfriend. I highly doubt that though. I don't think what Obito is seeing now is real, which also explains why he feels some sort of connection to what Kakashi sees, it may be Madaras dojutsu.
So the other possebility is that Rin died during the time skip, which is how Kakashi awakened his sharingan, but that would be risky for Madara since Obito might have met her during that period. Except if she was in ANBU doing lots of missions, which could explain it.
Maybe she and Kakashi did an ANBU mission during the time skip where she died, although I also doubt that.
In fact I'm not really sure what to think, but we will probably see in the next chapter. Might also be Kakashi just didn't become able to use the sharingan till after the time skip, but taking that he was a genius at the level of Itachis that would also be doubtful for me.
Maybe a fourth option, who knows..