How can I move my player on exact x and y coordinates, like on a grid?

Question

I'd like to implement movement exactly like it is done in this game. (You can see it in action on YouTube.)

I need my Player to always stay in the center of X and Y similar to grid movement, but with free movement/line like. Only move on x and y lines. And I need smooth changes at intersections.

 var x = Input.getAxis("Horizontal");
 var y = -Input.getAxis("Vertical");
 var deltaPos = Vector2.zero;

 if (Mathf.Abs(inputY) > Mathf.Abs(inputX)) {
     deltaPos.Y = speed * inputY * Time.deltaTime;
 }
 else {
     deltaPos.X = speed * inputX * Time.deltaTime;
 }

 transform.position += deltaPos;

Answer 1

Disabling the diagonal movement could be a good enough option. The example looks like they did that along with adding box colliders notice that the characters can be overlapped under the blocks at some points in the video. This may be enough to get you going on the right path.

void Update()
{
    TranslateNonDiagnalMovement(Input.GetAxis("Horizontal"), Input.GetAxis("Vertical"));
}
void TranslateNonDiagnalMovement(float x, float y)
{
    x = y > .1f | y < -.1f ? 0 : x;
    y = x > .1f | x < -.1f ? 0 : y;

    transform.Translate(x * 3, y * 3, 0);
}