The $d(n):=D(n,n)$ is OEIS sequence A005773. The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$. A proof could come from a similar proof of the Hankel determinant property of the Catalan numbers. A related property is to use $d_0=1$ and take $\det(H_n^0)$ with $d_0$ in the first row and column giving OEIS sequence A163806 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.
The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. This can be used to show that the unique solution to $\det(H_n^0)=\det(H_n^1)=1$ for all $n$ is the Catalan numbers. For a reference look at Aigner, Catalan-like numbers and determinants, J. Comb. Th. A 87 (1999), 33-51.
By the way, notice how the diagram for $D(4,4)=13$ is $$ \begin{matrix} 1 & 2 & 5 & 13 \\ 1 & 3 & 8 & 21 \\ 1 & 3 & 8 & 21 \\ 1 & 2 & 5 & 13 \end{matrix}$$ where each entry is the sum of two or three entries in the preceding column.
Catalan and related sequence proofs are given by J, Cigler in Some nice Hankel determinants.