Couldn't this be referring to some other type of similarity, rather than appearance?
I suppose so but that interpretation would be irregular and you’d have to specify.
"The same" is a much stronger comparison than "similar", so it would not be equivalent.
'You're not alike' is more natural in English than 'You are not similar' or even 'You are not similar to each other', which Duolingo accepted. 'You are alike' should be accepted, if it isn't already (I haven't tried giving it as an answer).
Agreed. "You are not alike" or "You are not similar to each other" are the proper English reanslations. Hebrew here is like Spanish, it has the verb "to be similar", but English doesn't.