No, NDT is right on this one.
Multiplicative identity applies to pure numbers. But when you're working with real-life objects, you have to include the units in your operation as well. You ALWAYS have to include your units when you are making calculations in physics.
A penny times 1 is denoted: 1 cent x 1 = 1 cent The unit of the first quantity is "cents", and there is no units on the second quantity, so the final units are still just "cents".
However, a penny times another penny is denoted: 1 cent x 1 cent = 1 cent^2 The unites of both quantities are "cents", so you have to multiply them together to give you "cents squared". And cents squared is meaningless, it doesn't exist, because you can't multiply a penny by another penny. It doesn't make any sense, there's no way to combine pennies via multiplication. You can add pennies together, and you can multiply pennies by any pure number you want (which in reality is the same as adding), but you can't multiply pennies by each other.
If you are working in physics it makes more sense. You can multiple velocity by time, and the units work out to give you distance. You can multiply mass by velocity, because those units give you momentum. You can multiply mass by acceleration, because that gives you force. You can multiply velocity by velocity, because that gives you acceleration, which is velocity squared. You can multiple distance by distance, because that gives you area, which is distance squared. You can even multiple distance by distance by distance, because that gives you distance cubed, which is volume.
But you can't multiply a penny by a penny. It's meaningless, pennies squared are not anything.