How do we know that two physical objects are exactly the same?

We can establish a way of measuring both objects based on something: molecules, wavelengths, distances, composition, design, etc. Each of these metrics gives us a quality check. We can go from fine to coarse-grained measurements. Fine-grained ones will provide us more confidence, usually at the cost of being more expensive to measure, while coarse-grained measurements will be cheaper but will have a more significant margin of error.

With those measurements, we compute the difference, and if the result is zero, we have the same object based on that measurement. 

Thus, I would say it is "difficult" to know if two physical objects are the same if you want great confidence. 

Examples at the molecular level would be DNA and at color wavelengths. 

Thus, can we have something that allows us to certify a measurement that we can challenge against the physical object to check if the measurement is exactly what the object is claiming?

Capture verifiably real moments in the age of AI

Absolutely! Can you elaborate on that further?