One was when he first awakened it and he was knocked out after. In that case it was obito who awakened it.
The second awakening isn't really an awakening because he had already awakened it and forgotten. In that case what tobirama said doesn't contradict at all because kakashi had it already due to obito, just forgotten about it.
Again this isn't the first time in an anime we have seen that a character's tool/weapon, etc awakens when they shows some form of emotion at their masters grief, even in death.
Their eyes are linked.
Ill give you examples.
Naruto as we know is kurama's jinchurcki. Yet when gin/kin bro's went bijuu mode, naruto could feel kurama's chakra [
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]. There is a form of link between them because of kurama.
Or zetsu reacting to naruto's chakra and turning into a tree, similar to yamato using mokuton.
These examples aren't exactly the best, but as you can see, because certain opponents share some trait/ability they might also have some form of connection, or perhaps even the jinchrucki realm that jins can enter