As @MaoWao pointed out, there is no spectral triple for this choice of $H$, but if we make $H$ larger, we can get a spectral triple.
This is an explicit construction in the complex case from section 3.2 of A dual formula for the spectral distance in noncommutative geometry by Francesco D'Andrea and Pierre Martinetti for $H := \mathbb C^N \otimes \mathbb C^m$ for $m \ge 1$: for Hermitian matrices $\left(L^{(k)}\right)_{k = 1}^{m} \subset \mathbb C^{N \times N}$ let
$$
D_L
:= \sum_{k = 1}^{m} L^{(k)} \otimes E_{kk}
\in \mathbb C^{N \times N} \otimes \mathbb C^{m \times m},
$$
where $E_{i, j} \in \mathbb C^{m \times m}$ for $i, j \in \{1, \ldots, m\}$ is the $(i, j)$-th matrix unit.
We identify the operator algebra $(B(H), \circ, *)$, where the multiplication $\circ$ is composition and the involution $*$ means taking the adjoint operator, with $\mathbb C^{N \times N} \otimes \mathbb C^{m \times m}$ by the same $*$-isomorphism used in the question applied to both factors of $H$ individually.
The C$^*$-algebra $\mathcal A := (\mathbb C^{N \times N}, \cdot, \ {}^{\mathsf{H}})$ with the usual matrix multiplication and the Hermitian adjoint as involution, acts on the first factor of $H$, which means the representation of $\mathcal A$ in $H$ is
$$
\pi \colon \mathcal A \to B(H), \qquad
A \mapsto \big( x \otimes y \mapsto (A x) \otimes y \big).
$$
Verification of the properties.
Indeed $\pi$ is a *-homomorphism: for $A, B \in \mathcal A$ we have
$$
\pi(A \cdot B)
= \big( x \otimes y \mapsto (A B x) \otimes y \big)
= \big( x \otimes y \mapsto (A x) \otimes y \big) \circ \big( x \times y) \mapsto (B x) \otimes y \big)
= \pi(a) \circ \pi(b)
$$
and
$$
\pi(A^*)
= \big( x \otimes y \mapsto (A^{\mathsf{H}} x) \otimes y \big)
$$
and $\langle \pi(A) h_1 \otimes h_2, k_1 \otimes k_2 \rangle_H = \langle h_1 \otimes h_2, \pi(A)^* (k_1 \otimes k_2) \rangle_H$ for all $h_1 \otimes h_2, k_1 \otimes k_2 \in H$, that is
\begin{align*}
\langle (A h_1) \otimes h_2, k_1 \otimes k_2 \rangle_H
& = \langle A h_1, k_1 \rangle_{\mathbb C^N} \cdot \langle h_2, k_2 \rangle_{\mathbb C^m} \\
& = \langle h_1, [\pi(A)^*(k_1, k_2)]_1 \rangle_{\mathbb C^N} \cdot
\langle h_2, [\pi(A)^*(k_1, k_2)]_2 \rangle_{\mathbb C^m} \\
& = \langle (h_1, h_2), \pi(A)^* (k_1, k_2) \rangle_H.
\end{align*}
Hence $\pi(A)^*(h \otimes k) = (A^{\mathsf H} h) \otimes k$ and thus $\pi(A^*) = \pi(A)^*$.
The operator $D_L$ is self-adjoint: for $h := h_1 \otimes h_2, j := j_1 \otimes k_2 \in \mathbb C^{N} \otimes \mathbb C^{m}$ we have
\begin{align*}
\langle D_L h, k \rangle_{H}
& = \left\langle \sum_{k = 1}^{m} (L^{(k)} h_1) \otimes (E_{kk} h_2), j \right\rangle_H \\
& = \left\langle \sum_{k = 1}^{m} (L^{(k)} h_1) \otimes (E_{kk} h_2), j \right\rangle_H \\
& = \sum_{k = 1}^{m} \left\langle (L^{(k)} h_1) \otimes (E_{kk} h_2), j \right\rangle_H \\
& = \sum_{k = 1}^{m} \langle L^{(k)} h_1, j_1 \rangle_{\mathbb C^N} \langle (E_{kk} h_2), j_2 \rangle_{\mathbb C^m} \\
& = \sum_{k = 1}^{m} \langle h_1, L^{(k)} j_1 \rangle_{\mathbb C^N} \langle h_2, E_{kk} j_2 \rangle_{\mathbb C^m} \\
& = \ldots = \langle h, D_L k \rangle_{H}.
\end{align*}
The spectral distance between the density matrices $p_1$ and $p_2$ induced by $D_L$ is equal to (proposition 5)
$$
d(p_1, p_2)
:= \inf\left\{ \text{tr}(\sqrt{u^* u}): u = (u_k)_{k = 1}^{m} \in \mathbb (C^{N \times N})^m, \sum_{k = 1}^{m} \big[ L^{(k)}, u_k \big] = \rho_1 - \rho_2 \right\},
$$
and the initial definition by Connes reads
$$
d(p_1, p_2)
= \sup\left\{ \big| \text{tr}\big(A (p_1 - p_2)\big) \big|: A \in \mathcal A, \ \| [D_L, \pi(A) ] \|_H \le 1 \right\},
$$
where
\begin{align}
\| [D_L, \pi(A) ] \|_H
& = \| D_L \pi(A) - \pi(A) D_L \|_H \\
& = \left\| \sum_{k = 1}^{m} (L^{(k)} A) \otimes E_{kk} - (A L^{(k)}) \otimes E_{kk} \right\|_H \\
& = \left\| \sum_{k = 1}^{m} [L^{(k)}, A] \otimes E_{kk} \right\|_H \\
& = \sqrt{\left\langle \sum_{k = 1}^{m} [L^{(k)}, A] \otimes E_{kk}, \sum_{j = 1}^{m} [L^{(j)}, A] \otimes E_{jj} \right\rangle_H} \\
& = \sqrt{\sum_{j = 1}^{m} \sum_{k = 1}^{m} \left\langle [L^{(k)}, A], [L^{(j)}, A] \right\rangle_{\mathbb C^{N \times N}} \cdot \left\langle E_{kk}, E_{jj} \right\rangle_{\mathbb C^{m \times m}}} \\
& = \sqrt{\sum_{k = 1}^{m} \left\langle [L^{(k)}, A], [L^{(k)}, A] \right\rangle_{\mathbb C^{N \times N}}} \\
& = \sqrt{\sum_{k = 1}^{m} \left\| [L^{(k)}, A] \right\|_{\mathbb C^{N \times N}}^2},
\end{align}
as
$$
D_L \pi(A)
= \sum_{k = 1}^{m} (L^{(k)} \otimes E_{kk}) \cdot (A \otimes \text{id}_m)
= \sum_{k = 1}^{m} (L^{(k)} A) \otimes E_{kk}.
$$
