Speaking personally, I’m not sure I understand that thinking about the true/false result feeling wrong. It seems like you’re taking for granted that there’s a formal concept of “identicality” which two values with the same abstract representation must have. I would say that that concept is real, but we are deciding a name for it. The suggestion here is to call it “trivially identical” for clarity and deliberately leave “identical” semantically uninhabited.
That is, you are parsing it as “is (trivially (identical))”, with all of the semantics being borne by “identical” and “trivial” just being a description of how the check is done, when you should instead parse it as “is (trivially identical)”, where the semantics are borne by the two words in combination.
This is not particularly unusual. Off the top of my head, there is a property of types where they can be “fully inhabited”, meaning that all bitwise representations are valid values of the type. This should be understood as its own formal property, not just a description of one way of having the “inhabited” property.
It’s nice that “is trivially identical” can be read largely correctly the way you’re reading it, but I think it’s fine to say that, if someone wants to be quite pedantic, it should really be read in this other way.