There are no local obstructions for constant distinct singular values when $M$ and $N$ have dimension $2$. This is locally a determined symmetric hyperbolic system of two equations for two unknowns, so it's always locally solvable.
Note 1: Local character of the equations
Giving a general analysis in all dimensions in classical PDE terms is not so easy notationally, although, for example, computing the linearization of the equations in the flat case at the particular solution
$$
f(x^1,\ldots,x^n) = (\lambda_1 x^1,\ \ldots,\ \lambda_n x^n)
\tag1
$$
where $0<\lambda_1<\lambda_2<\cdots<\lambda_n$ and seeing that this linearization is the system of $n$ equations
$$
\frac{\partial u^1}{\partial x^1} =
\frac{\partial u^2}{\partial x^2} = \cdots
= \frac{\partial u^n}{\partial x^n} = 0
\tag2
$$
for $u = \bigl(u^1(x),\ldots, u^n(x)\bigr)$ is not hard.
For general metrics in $n$-dimensions, linearizing at a particular solution can be a bit messy, but, given a solution $f$ in a neighborhood of a point $p\in M$, one can always introduce $p$-centered coordinates $x^i$ and $f(p)$-centered coordinates $y^i$ on a neighborhood of $f(p)\in N$ so that, in these coordinates, $f$ near $p$ is of the form $(1)$ and so that the constant singular values PDE system for maps $u$ that are $C^1$-close to $f$ can be solved in the form
$$
\frac{\partial u^i}{\partial x^i} = E^i\left(x,u,\left[\frac{\partial u^j}{\partial x^k}\right]_{j\not=k}\right)\tag3
$$
where the functions $E^i$ are smooth and vanish to second order at the origin. Thus, the linearization at $u=f$ will be of the form $(2)$. The point is that the coefficients of the actual metrics on $M$ and $N$ do not materially affect the form of the linearization.
Now, the system $(3)$ can be put in Cauchy form, and a solution $u$ near $p$ is completely determined by its restriction to a non-characteristic hypersurface $H^{n-1}\subset M$ passing through $p$, i.e., one whose conormal $\xi_1\,\mathrm{d}x^1 + \cdots + \xi_n\,\mathrm{d}x^n = 0$ at $p$ has $\xi_1\xi_2\cdots\xi_n\not=0$. Thus, the solutions depend locally on $n$ functions of $n{-}1$ variables, in the sense of Cauchy-Kowalewskaya.
When $n=2$, the fact that the characteristics are real and distinct implies that, in fact, the Cauchy problem is locally solvable in the smooth category. This result may generalize to $n>2$, but I'm not sure about that without checking the literature. In the real-analytic category, though, local solvability of the non-characteristic initial value problem immediately follows from the theorem of Cauchy-Kowalewskaya.
Note 2: The initial value problem
Here is what the above analysis, together with a little geometry, says about the initial value problem: Fix Riemannian $n$-manifolds $(M^n,g)$ and $(N^n,h)$ and an increasing sequence of positive constants: $0<\lambda_1 <\lambda_2<\cdots <\lambda_n$. Now let $H^{n-1}\subset M$ be a simply-connected smooth embedded hypersurface in $M$ and suppose that one can find a smooth map $\phi:H\to N$ such that the (not necessarily constant) singular values of $\phi':TH\to TN$ are smooth functions $\mu_i:H\to \mathbb{R}$ that satisfy
$\lambda_i < \mu_i < \lambda_{i+1}$ for $1\le i < n$. Then one can show, by linear algebra that, for each $p\in H$ there exists $2^n$ choices for a linear map $\psi:T_pM\to T_{\phi(p)}N$ such that $\psi(v) = \phi'(v)$ for $v\in T_pH$ and the singular values of $\psi$ are $\lambda_1,\ldots,\lambda_n$. As $p\in H$ varies, these $2^n$ choices form a smooth covering space of degree $2^n$ over $H$. Since $H$ is simply-connected, it follows that one can choose a smooth isomorphism of vector bundles $\Phi: H^*(TM)\to \phi^*(TN)$ such that $\Phi(v) = \phi'(v)$ for $v\in TH\subset H^*(TM)$ and, for each $p\in H$, $\Phi:T_pM\to T_{\phi(p)}N$ has singular values $\lambda_1,\ldots,\lambda_n$. (There are $2^n$ such choices for $\Phi$).
Then the Cauchy-Kowalewskaya theorem says that, in the analytic category, there exists an open neighborhood $U$ of $H$ in $M$ and an analytic map $f:U\to N$ such that $f(p) = \phi(p)$ and $f'(p) = \Phi(p)$ for $p\in H$ and the singular values of $f$ are constant and equal to $\{\lambda_1,\ldots,\lambda_n\}$.
When $n=2$, this existence theorem holds in the smooth category. It might hold in the smooth category for all $n$, I don't know about that, but probably someone such as Deane Yang or Dennis DeTurck could determine this.
In any case, even in the smooth category, the existence result holds formally in the sense that, for each integer $k\ge 0$, there exists a smooth map $f_k:U\to N$ such that $f_k(p) = \phi(p)$ and $f_k'(p) = \Phi(p)$ and the singular values $\sigma_1,\ldots,\sigma_n$ of $f_k$ (which are not necessarily constant) have all their derivatives up to order $k$ vanish at $p\in H$, and this condition uniquely specifies the derivatives of $f_k$ up to order $k{+}1$. In particular, there is no local pointwise obstruction to the existence of an extension $f$ of $\phi$ that has constant singular values $\{\lambda_1,\ldots,\lambda_n\}$, since one can find an approximate solution (in fact, $2^n$ of them) to any desired order.
