If the metric is defined on a bounded subset of the x-y plane,let's say a closed square area $0\le x,y\le1 $, the metric is defined as $$\langle u,v\rangle =\langle (u_x,u_y),(v_x,v_y)\rangle =\langle u_x,v_x\rangle +w\langle u_y,v_y\rangle $$ where $u_x,u_y$ are the components of the tangent vector in $x,y$ directions respectively,$w$ is a weighting factor. For the case $w\rightarrow\infty$, is the manifold defined by such a metric Riemannian?
My question is: if it's a Riemannian manifold, then it should have a finite diameter since $0\le x,y\le1$ is compact. But obviously there are points with an infinite distance between them, for example the point pair $[(0.1,0.1),(0.5,0.5)]$ . Something wrong with my configuration? Or it's a kind of degenerated structure?
It makes sense to consider a family of Riemannian manifolds $M_w$ with the weight factor $w$, and investigate their limit as $w\to\infty$ (usually, with respect to some form of the Gromov-Hausdorff metric, but there are other options too). The limit may be an interesting object, or not: depends on how you deform the metric with $w$.
In the case you described, the space $M_w$ is a rectangle of size $1\times w$. These don't converge as $\omega\to\infty$ in the standard GH sense, but they do converge in the pointed GH sense. The limit is an infinite strip, $[0,1]\times \mathbb{R}$.
There is a better known construction of this kind: approximation of a sub-Riemannian manifold with Riemannian manifolds. The principle is the same: tangent vectors in certain directions ("horizontal" subbundle of the tangent bundle) are given their Euclidean length, the other directions are penalized by a large factor $w$. The difference is that in the sub-Riemannian case, the horizontal subbundle is rich enough so that any two points can be connected by a curve tangent to it. Therefore, for any two fixed points $p,q$ the distance $d_w(p,q)$ has a finite limit as $w\to\infty$; namely the shortest length of a "horizontal" curve connection $p$ to $q$.
In your case, the "horizontal" vectors are literally horizontal, and as you observed, one cannot connect two arbitrary points moving in the horizontal direction.