For interior doors, you can do that for soundproofing, but usually the door itself is already a weak point. And with ventilation systems, the whole thing is counterproductive
In terms of sound, the seal is, in my opinion, the first effective measure. Additionally, you can then make the entire door sound-excited. Conversely, a sound-excited door without a drop seal is, in my opinion, not very useful.
In connection with ventilation systems, there are also sound-damping overflow valves or you guide the sound over the door frame or use the overflow drop seal in the door (the air is redirected twice here, hence the soundproofing while the door is still sealed to the floor). Suitable search terms would be, for example, "soundproof overflow element"; "soundproof inflow valve"; "Planet drop seal Mine-F" ...