Why does a change of direction imply an acceleration? We know that it takes no energy to change the direction of a vector, we know also that it takes no energy to displace a body in motion if a push is applied exactly at a right angle on its center of mass.
Considering that, if that is true, can you explain why a simple change of direction without any increase of speed is considered an acceleration?
 A: Answer to the question in the title?
Two vectors are only equal to each other if they are the same (this is a general rule: equality means the things compared are identical). That means having the same direction as well as the same magnitude.
So how could changing the direction of motion not be acceleration?
Don't get hung up on fact that in 1 dimension acceleration always involved changing the magnitude, just ask if the final and initial velocities are the same or not.
Comment on the body text
In the body you ask about kinetic energy. There are two issues that come up with that:

*

*Kinetic energy (or any energy) is a scalar, and so it can remain the same while the velocity changes (as in uniform circular motion).


*Changes in (or differences of) velocity are the same for all inertial observer, but changes in kinetic energy are not. If you are thinking of the work-energy theorem you'll notice that this makes the work done frame-dependent as well. But this is not a surprise because the length and direction of the path is frame-dependent.
A: This becomes easier to understand if you think of velocity being made up of perpendicular components.  
For example, let $v = v_x\hat i + v_y\hat j$.
That is, velocity is made up of an x-component, $v_x$ and a y-component, $v_y$.
When there is a change of direction, the $v_x$ and $v_y$ components will change.  This means there must be some horizontal acceleration and some vertical acceleration.  Overall the magnitude of the velocity might not change, but you will still need horizontal and vertical acceleration to change the values of $v_x$ and $v_y$ which is required for a change of direction.
A: Very briefly said:


*

*A change of velocity $\mathbf{v}$ in time  means that $\frac{\textrm{d}\mathbf{v}}{\textrm{d}t} \neq \mathbf{0}$ 

*The acceleration $\mathbf{a}$ is defined as $\mathbf{a}=\frac{\textrm{d}\mathbf{v}}{\textrm{d}t}$

*Therefore $\mathbf{a} \neq \mathbf{0}$ or in words if your velocity changes you will have an acceleration. Velocity also changes when only its direction changes -> that is why this is considered an acceleration 


Nowhere here do you have to apply any considerations about energy.
A: Your premise about conservation of energy is correct: A change in direction does not require energy. The applied force and the direction of distance travelled are perpendicular, and thus the work applied W = F.s will be zero, since both vectors are perpendicular. 
The acceleration is not zero, since there is deceleration in the x direction, and acceleration in the y direction.
