The Gram matrix depends on the choice of a lattice basis (it's just the matrix of inner products of the basis vectors), which means all lattices in two or more dimensions have infinitely many Gram matrices, typically with very different eigenvalues. For example, for the integer lattice $\mathbb{Z}^2$, the basis $(1,0)$, $(0,1)$ has the identity matrix as its Gram matrix, while the basis $(2,3)$, $(3,4)$ has the Gram matrix $\begin{pmatrix} 13 & 18\\ 18 & 25 \end{pmatrix}$, which has eigenvalues $0.02633403\dots$ and $37.97366596\dots$. (To see that $(2,3)$ and $(3,4)$ span $\mathbb{Z}^2$, note that they certainly span a sublattice, and it can't be a proper sublattice since the basis matrix has determinant $-1$.) There's some regularity; for example, the determinant of the Gram matrix of a basis depends only on the lattice. However, the eigenvalues do not.

The Gram matrix uniquely determines the lattice basis, up to orthogonal transformations (i.e., isometries). That means the three given bases of $E_8$ are genuinely different from each other, but they all span the same lattice. This follows directly from known characterizations of $E_8$ (for example, it's the unique even unimodular lattice in $\mathbb{R}^8$), and it can also be checked by identifying the correct integer linear transformations to express each basis in terms of the others (see below for computational details).

So the issue here is that the Gram matrix is an invariant of the basis, not the lattice, and there are infinitely many different bases. I don't think the number $3$ has any particular significance here. It's just the number of bases given in Nebe and Sloane's catalog.

I don't know of any special reason why $E_8$ has more different bases listed by Nebe and Sloane than other lattices do. I'm not surprised that it has more than most lattices, because it's a particularly important lattice that can be constructed in a variety of different ways. However, I would have guessed that some other lattices might have as many.

To compute the change of basis matrices, one can use PARI/GP (see https://pari.math.u-bordeaux.fr). For example, two of the Gram matrices listed for $E_8$ by Nebe and Sloane are $$G_1 = \begin{pmatrix}
4 & -2 & 0 & 0 & 0 & 0 & 0 & 1\\
-2 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 2
\end{pmatrix}$$ and
$$G_2 = \begin{pmatrix}
2 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & 2 & 0 & -1 & 0 & 0 & 0 & 0\\
-1 & 0 & 2 & -1 & 0 & 0 & 0 & 0\\
0 & -1 & -1 & 2 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 2
\end{pmatrix}.$$
Suppose we give them to gp as variables G1 and G2. Then the command qfisom(G1,G2) will give us a matrix
$$
A = \begin{pmatrix}
0&0&0&0&1&-2&0&-1\\
0&0&0&1&0&-2&0&-1\\
0&0&0&1&0&-3&0&-2\\
0&0&0&2&0&-4&-1&-2\\
0&0&0&2&0&-4&0&-1\\
1&0&0&1&0&-3&0&0\\
0&0&1&0&0&-2&0&0\\
-1&1&0&0&0&-1&0&0
\end{pmatrix}
$$
such that $A^t G_2 A = G_1$. What this means is that $A$ is the change of basis matrix between these bases: the columns of $A$ tell which linear combinations of the basis vectors corresponding to $G_2$ give the basis vectors corresponding to $G_1$. In other words, if the basis matrix $B_i$ has the basis vectors as columns, then $G_i = B_i^t B_i$ and $B_1 = B_2 A$.