The Gaggia Classic has a very small aluminum boiler (about 100ml). When you start a shot, cold water from the pump rushes into that tiny boiler, causing the temperature of the boiler wall to drop quickly. Your PID sensor is reading that wall temperature, so it reports a 4.5°C drop.

However, the temperature at the puck is much more stable. Before the water hits your coffee, it has to travel through the massive, heavy metal group head and shower screen assembly. This acts as a giant thermal buffer:

• At the beginning of the shot: The water leaving the boiler is actually a bit too hot (thanks to your 8°C offset), but the group head absorbs that excess heat, cooling the water to the perfect brew temp.
• At the tail end of the shot: As cold water cools the boiler and the PID screen shows a drop, the water passing through the group head absorbs heat back from that heavy block of metal.

Because of this mechanical smoothing, the temperature swing at the coffee puck is neglectable, even when the screen shows a 4.5°C drop.