Glazing Unit in New Composite Door Moves

Recently got a new front door - the Endurance Fuji door which has a narrow pane of glass in the middle of the door.

Is it normal for the glass unit in the door to have a slight movement to it when its pushed?

The reason I pushed it was because I can hear a small amount of outside noise coming through the door, I went around the door and listened carefully to see if I could find the area where the noise was coming through. My ear led me to the glass unit in the centre of the door - I then thought the sound might be coming through via the sides of the glass... so I pushed on the glass unit to see if it moved a little, and it does, a very small movement which doesn't happen when I push on the glass in the side panels.

Is this normal or is it a bad installation?

no expert
but you will have a bit off movement
the rubber gasget or seal should fill the gap stopping air movement and giving an air tight seal

Out of curiosity (I like to learn) why would there need to be movement for the glazing on a composite door but not on the side panels?

side panel will tend to be narrower than the doors
so smaller movements in the glass or structure
all materials move will expansion and contraction
but in general will not move to a noticeable extent
rubber gasgets will tend to absorb any movement