"Distance" in the AAD vs the SD

I've heard people describe the difference between the average absolute deviation and the standard deviation as the average distance from each data point to the mean, where one uses Manhattan distance, and the other Euclidean. I heard it's because one converts to absolute value, and the other squares then takes the square root, but the explanation doesn't really work for me

What are the axes? Does each deviation (let's say n) get its own dimension? Are we really just building a triangle-pyramid whatever thing in n dimensions in euclidean space and either solving for the average side length for AAD or solving for the length of a vector between point (0,0,0...) and point (|n1-E(X)|,|n2-E(X)|,|n3-E(X)|...) divided by n for SD?