I'm trying to gain a non-trivial understanding of the Law of Identity, in Logic
The Law of Identity states that each object is identical to itself -- hard to deny. "Daniel is identical to Daniel" is a particular instance of that Law.
Your clone is not identical to you: if you and your clone we're alone in a room and we counted the number of objects in the room, we'd get two, not one.
"Daniel is identical to Daniel" does not express that the word to the left of "is identical to" is the same word as the word to the right of it: it expresses that the object the first word refers to is the same as the object the second one refers to. This can be made plainer by considering, for instance, this claim: "Daniel Defoe is identical to the author of Robinson Crusoe." This is a true identity claim, even though the words "Daniel Defoe" and "the author of Robinson Crusoe" are not themselves identical.
It would be difficult to say why the Law of Identity is true. Any defense of it would either involve using words like "equals," "same as," etc. all over again, or would use words that are themselves less clear than the word "identical." For that reason, I wouldn't understand anyone who questioned the Law. If someone said that Bob Dylan wasn't identical to Robert Zimmerman, I'd simply assume he wasn't using "identical" to express the same relation mentioned in the Law of identity.