Saying "It may be different with each family,..." means that "it" varies among families.
Does it mean that using "different with" doesn't mean anything like "comparison"?
No. It is more like it varies with different families.
It can't be different if it is not compared to anything else. Your dictionary is right. It is different (from the hypothesis or sample in question) with each family.
Let's see if this helps, Little Mon You. When people use the pronoun "it" to mean "something","that thing", they generally use "with" when they use the word with "different":
Carl likes jazz. It's different with me. I like rock.
I am different from Carl. The music that I like is different from the music that Carl likes.
Does that help?
No, you cannot say different with each other It can only be different from each other. Different with and different from have two distinct meanings and cannot work interchangeably.