Could different views of the universe simply be different points in time?
Could this occur if the distant body is moving laterally at a fast enough velocity relative to our position


Nope. I mean if the universe is flat it doesnt matter how fast the object goes you cannot see it twice in the sky. 


  or if the shape of the universe itself loops around?


Yes. If the universe has a positive curvature. When we look at the sky, we can see the same object in two opposite directions. 

In our universe this is not possible since if the universe is positively curved then the radius of curvature is much larger then the size of the observable universe. So its highly unlikely that we will see the same object in two different positions in the sky.