This is a tricky question.

1. First off, the straight line by itself is not distance. Its length is.
   
2. Second, when you say "shortest", what it is you compare it with? In common systems of axioms, the triangle inequality is proved. But,
   
3. it takes means beyond plain geometry to define a curve, let alone its length.
   

>Is it
>same in the 3- dimensional space?

Yes.

In analytic geometry the triangle inquality is an algebraic fact. In more general spaces the triangle inequality is part and parcel of the definition of the distance function.