Move objects with mouse while paused (2D)

How would I be able to make specific objects in the game world (marked with its own material) move using the mouse while in game?

Any drag and drop code you find (and it has been posted to UA many times), that use OnMouseDown() rather than raycasting will work in 2D. Raycasting solutions would need a bit of change for 2D. Here is one link. The script will need to be attached to each object you want to be draggable:

Note if you only want to only have it draggable if the game is paused, then you can add something like this to the top of the OnMouseDown() callback:

if (Time.timeScale > 0.01) return;