The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load
Thevenin equivalent circuit is a way to represent a whole and relatively complex circuit into a simplest circuit diagram which give you same voltage between the nodes,that you use as reference to calculate the thevenin circuit. So you can see , it only fulfil one condition, it will give you same voltage at the nodes which you would get from main circuit and so current through the nodes will depend on the load you put between those two node.

You can not get any data about what would happen inside the main circuit and it's internal nodes.

So you can easily understand that  physically only nodes across the load is only two node which actually exist in the main circuit also. So if you wanna get I-V characteristics from thevenin circuit, you have no other choice but to use nodes across the load to collect data which represent data of main circuit.

FYI: I write this as far as I understand but It may be wrong.anyone is welcome to find my mistake.