After pasting an object I make sure only one is selected, but both the original and the copy move together. I can enter edit mode and move one but they remain joined and ctrl+p to separate does not work. I'm not sure what has changed but I used to be able to copy an object and paste it as an entirely separate object.
I had the same issue today and I had found the solution.
NOTE: This is not a permanent solution, you would have to do this every time you would paste an object in the project file.
Once you start to move your pasted object, there will be a tab called 'MOVE'. There you will the selection of options of how you want to move the object.
Under the tab, there will be an option called 'Proportional Editing'. You would want to turn that off.
Then you will be able to freely move your pasted object without interfering with the original object.
Hopes this helps!