"I mistook Emily for her sister last night
Emily was mistaken."
In this context "mistook" = "was mistaken". Mistook (mistaken) means a mistake, it does not mean misunderstood. You made the mistake, not Emily.
You could also have written "I was mistaken in thinking Emily was her sister last night."
Your second statement is wrong because it contradicts the first one. You said "I mistook..." (or "I was mistaken...") then you said "Emily was mistaken." She wasn't mistaken, you were.
If you wanted to specify that Emily made her own mistake, you would have needed to write more information. For example, "Emily was mistaken in thinking you knew what she looked like."
or
"Emily mistook you for someone observant."
I mistook the idea of leadership.
The idea was mistaken. (It was misunderstood)
Similar here. The idea did not make a mistake, you did.
Therefore, the sentence 'I guess Julia was mistaken' doesn't mean that Julia was wrong in her opinion, it means that she was misunderstood by someone. Wasn't she?
In fact the sentence 'I guess Julia was mistaken' doesn't mean that Julia was wrong in her opinion' does mean Julia was wrong. Everyone understood her, but she was wrong.
There are also lines from the song by cranberries:
'Who are we mistaken?'
A song is not a good source of information. Language can be changed to make it singable and lyrics can be disputed. The line is actually "Who are we mistaking?" I believe this is referring to not making an error in judgment towards anyone because in fact there is war.
One more example:
she'd been mistaken about his intentions. It's clear that she had got wrong his intentions.
But in terms of passive voice, it would be 'someone's intentions had been mistaken' They were misunderstood.
Indeed someone had been mistaken, she had been. She had been mistaken in her judgment towards his intentions. It was "she" who was wrong, not the intentions. She is the subject.
Could someone explain what did I get wrong?