I can understand that objects floating on water follow that principle but that's because they are floating. Once fully suspended in water, I think the amount of water they displace is just their volume. Shouldn't it be likewise for aircrafts in the atmosphere?
Not quite. Outside of ground effect, a downward force is applied to the air mass around the aircraft, resulting in lift on the aircraft. IIUC, this results in compressed, high pressure air below the plane and low pressure air above it. The result of that is a pair of vortices roughly centered on the plane's wingtips. These vortices themselves are Somehow Important to the whole thing. Anyway, you can see the deal when a plane flies close to a cloud top.[1]
In ground effect (i.e. the aircraft is flying within roughly one wingspan of the ground), the proximity of the ground blocks the formation of the wingtip vortices (?) and greatly enhances the efficiency of lift production (by some form of magic, AFAIK).
[1] https://s-media-cache-ak0.pinimg.com/564x/1c/33/e2/1c33e2f2f...
The only way mass would have an effect is if the exhaust (which heavier planes presumably generate more of) contributed to the sonic boom, which doesn't sound right.
I'll believe that modern planes might generate smaller sonic booms, being more efficient sounds like it's related to how you displace the air, which sounds very related to the size of the boom, but I don't believe you that mass is related.