This is not really an answer, but rather a longish comment and a suggestion about how one might focus the question a bit better.

First, when one asks for a 'canonical' isometric embedding into some Euclidean space of a genus $2$ surface endowed with a metric of curvature $K=-1$, one might want to consider in what sense even the Clifford torus is 'canonical'.

In fact, there are many flat metrics on the 1-holed torus: If $\Lambda\subset\mathbb{C}$ is a lattice (i.e., a discrete subgroup of rank $2$, say generated by $1$ and $\tau\in\mathbb{C}$ with positive imaginary part), then the standard metric $\mathrm{d}z\circ\mathrm{d}\bar z$ induces a flat metric $g_\Lambda$ on $T = \mathbb{C}/\Lambda$, and these flat metrics are not globally isometric as $\tau$ varies. Nevertheless, as Montiel and Ross showed in this paper, there is a $g_\Lambda$-isometric immersion $f_\Lambda:T\to\mathbb{E}^6$ by first eigenfunctions (that actually maps into a round $5$-sphere) that, moreover, is equivariant with respect to the identity component of the isometry group $G_\Lambda$ of $g_\Lambda$ in the sense that there is an embedding $G_\Lambda\to\mathrm{SO}(6)$ so that $f_\Lambda(g\cdot p) = g\cdot f_\Lambda(p)$ for all $p\in T$. In particular, this isometric embedding is as 'homogeneous' as possible. When $\tau=i$ (so that $\Lambda$ is the square lattice), this reduces to the Clifford torus example that the OP listed.

Of course, these examples are simply quotients of global *equivariant* isometric immersions of $\mathbb{E}^2$ into $\mathbb{E}^n$ for $n\ge 2$.

Meanwhile, it is well-known that the isometry group of the Poincaré disk is isomorphic to $\mathrm{PSL}(2,\mathbb{R})$ and that this group does not have any nontrivial homomorphism to the isometry group of Euclidean space in any (finite) dimension. Thus, there is no hope to construct an equivariant isometric immersion of the Poincaré disk into any (finite dimensional) Euclidean space, much less to find one that 'closes up' under some Fuchsian subgroup to give a 'canonical' isometric embedding of the $2$-holed torus into Euclidean space.

Thus, one must look elsewhere for a notion of 'canonical isometric embedding'. One possibility one might try to generalize the above flat case would be to ask for an isometric embedding by the eigenfunctions of the Laplace operator associated to a particular eigenvalue. A related (non-existence result) is the following: If $(M^2,ds^2)$ is a surface of constant negative Gauss curvature then there is no nontrivial map $f:M\to S^n\subset\mathbb{E}^{n+1}$ that satisfies $\Delta f = -\lambda f$ for any $\lambda\not=0$. This follows by the same techniques used to prove Theorem 2.3 in this paper of mine. Thus, some other kind of extra system of equations would be needed to determine what you mean by 'canonical'.