You are thinking of analog triggers, which work fine over Bluetooth. Adaptive triggers is game-controlled resistance and feedback within the trigger. A rifle trigger, for example, is light at first, then when you get close to the firing point it gets heavy for a moment and then ‘breaks’, suddenly going all the way.
This works on PC if you plug in via USB, but very few games support it on PC and Stam Input breaks it.
Haptics are high-quality rumble. Rumble is basically just turning a motor on and off, often in a high and a low mode. Modern controllers drive vibration motors at varying levels using a sound signal, giving far more nuanced control. They often also have more motors that are independently controlled. This also works when plugged in via USB.
These are often two components stacked. There’s a cabinet box and a shelf unit. They may not have attached them if it’s a temporary backdrop just for this video. It’s also possible that the shelf is bolted to the wall, allowing the cabinet to slide underneath it.