It makes sense that people, and all animals for that matter, can sense acceleration, as we live in a dynamic world, and acceleration, which is the rate of change of velocity, may be thought of as the forecaster for motion. If one can detect one's own acceleration, then one can accurately predict one's future position (provided one knows one's own current position and velocity with accuracy).
The major confusion among students when it comes to the study of motion is distinguishing acceleration from velocity. When my daughter asked if we were going down, she was asking if our velocity was pointing down. As the elevator neared its destination, it, and its contents, experienced a negative acceleration (braking), which resulted in a decrease in velocity. However, its velocity was upward throughout the deceleration: the elevator moved up, and slowed to rest.
The body can detect acceleration through a number of avenues: the muscles that maintain its configuration, its internal organs, its ears... Amazingly, the body cannot sense velocity. As one moves along in any vehicle with constant velocity (uniform motion), the only way one can detect the direction of one's motion is with one's eyes. If you wake up on a moving train with no windows, you have no way of determining which way the train is moving. As we move inside a vehicle, we glance out the window, note the relative velocity of the world outside the window, and state with confidence that we are moving the other way.
We can be fooled in a number of ways. Imagine you are on a large boat, and you judge your kinematics based on another large boat. We may deduce that we are moving based on the relative motion of the other large boat, only to be surprised when a stationary body of land is revealed behind it. We were still all along.
At Islands of Adventures, a theme park in Orlando, Florida, there is a superb ride that takes full advantage of our inability to correctly discern velocity: "The Amazing Adventures of Spider-Man". The entire ride takes place within a fairly small two-storey warehouse. Yet, the rider is under the impression that he or she is swinging twenty storeys in the air along with Spidey.
The trick is the multiple projection screens that are scattered within the warehouse. As the ride moves along, the rider is convinced that high-speed up and down motion is taking place simply because of the high-speed down and up motion of the optically projected background. The illusion is completed by a brief upward and downward acceleration (jolt) that bookend an ascent (the opposite for a descent). The acceleration is real, and is sensed. The velocity is virtual; it is faked. I suppose Spidey senses would render us less easily fooled.
This brings us back to the elevator. We can correctly assess our kinematics as we go from the ground floor to the eighth, and most adults do so without thinking about it. There are three phases of motion that take place here: acceleration, cruise, and deceleration.
Aware that the elevator starts at the bottom and from rest, the brief acceleration (felt because we feel slightly heavier as it takes place) implies that once it is over, an upward velocity is established. No acceleration for the long interval that follows (cruise) implies that this upward velocity is maintained. As we reach the eighth floor, we feel a bit lighter; we note that a downward acceleration is taking place. Since the elevator begins this downward acceleration with an upward initial velocity (that with which it had been cruising with), its speed decreases until it reaches its final destination, and returns to rest.
Note that speed and velocity are not the same thing. Velocity is defined by a speed and a direction. Since acceleration occurs any time a velocity is changing, it is possible to accelerate with constant speed: this happens when direction alone is changing. Consider a car that drives in a circle at constant speed (this type of kinematics is known as uniform circular motion). The passengers feel the same type of sensation that they feel when the car starts from rest and speeds up in a straight line. They feel they are being pushed opposite the direction of acceleration; they are simply feeling the effect of their respective inertias.
The notion of a windowless elevator is also the starting point for one of the centrepieces of modern physics. Albert Einstein was startled by the notion that one cannot decipher between acceleration and gravitation while encased inside the cabin of a windowless elevator. This idea was the catalyst for the theory that would become his masterpiece: that of general relativity.
A ride along an elevator is a nice opportunity to brush up on one's feel for classical physics. Maybe that is why kids like to press all of the buttons.