I edited that part. But whether you include "same" or not it doesn't change. The "same" is not directed towards the content of what kind of like/love it is. It's directed towards the word itself. (In the word love there is different content-family love, romantic love, friendship love, etc.) So I didn't put in the word "same" because doing so will ruin the context. To be more detailed, Kakashi's line can be translated to "even if you call it the same (word)love/like, the type of love/like is different."
Does any Japanese speaker have a different opinion? I can edit.
I honestly think it was just imported from mangapanda. The translations are pretty much the same. I don't think they are the raw version.
yeah that's what i thought about the 'same' used in that sentence, you explained it nicely here.