We walk blocks, not as the bird flies.
The Manhattan and Euclidean distance on finite spaces both generalize to "l_p norms" (1 and 2, respectively) on an n-dimensional Euclidean space, where ||x||_p = (|x_0|^p + |x_1|^p + ... + |x_n|^p)^(1/p). You can even naturally define it in the limit where p goes to infinity!