See previous comment. There is a physical relationship between inputs and outputs, and then there is a plethora of mathematical (and other) relationships which can be mapped onto the physical relationship.

What you're not grasping is that there are relationships which cannot be mapped onto the inputs and outputs. The circuit determines what relationships are possible. If it constrains them in certain ways, then the behavior of the circuit will be compatible with the laws of addition, and we can say that it implements those laws.

One may as well say that the words in a natural language intrinsically have certain meanings. If that were true, it would literally be impossible to utilize them in some inverted or nonstandard way, which is false.

Totally irrelevant. If I take a normal, 'functional' calculator, and interpret the symbols it produces in a way contrary to convention, that doesn't change the nature of the circuit within it. It still produces outputs whose relationship to the inputs implement the rules of addition. The meaning of the symbols comes from their use, and the circuit uses them in a particular way. If we assume that the calculator is consistent in its behavior, I could examine the relationships between the symbols it displays and determine what they mean, determine in what ways my interpretation fails to encompass the nature of the circuit's rules.