So why isn't, "They're nothing like me is", correct?

There are two actual grammatical rules here, one of which is slowly falling out of the language.

The first is subject vs object. "I" is used in the subject; "me" is used in the object.

The second rule is that the verb "to be" is special because it joins two things symmetrically.

"He is I" means the same as "I am he". Both "I" and "he" are subjects.

Your fundamental "rule" isn't any "rule" at all. "See if it makes sense" is bogus. You're relying on someone already having intuition about the language, and if that's wrong, they will get wrong answers from your "rule".

No claps.