Two lattices $\Lambda$ and $\Omega$ (the $\mathbb{Z}$-span of a linearly independent set $B\subset \mathbb{R}^n$) are said to be similar if there exist a real orthogonal $n \times n$ matrix $A$ and a nonzero constant $\alpha$ such that $\Lambda=\alpha A\Omega$. That is, if $\Lambda$ can be obtained from $\Omega$ through dilation and/or rotation.
Of course, having the same kissing number does not imply similarity by itself (there exist infinite classes of lattices $\Lambda \subset \mathbb{R}^2$ with $\lvert S(\Lambda)\rvert=4$, for example). This can be verified in https://arxiv.org/abs/1101.4442.
However, if they are also equally dense, can you say that they are similar? For example, is it possible to find a lattice $\Lambda$ not similar to $D_6$ but with $\kappa(\Lambda)=60$ and $\delta(\Lambda)=\tfrac{1}{2^4}= 0.0625$? Does this property hold true for $A_n$ and $D_n$, for example? Or in general?
Edit: the proven densest lattices in dimension 1 to 8, and 24, are known to be unique (Conway&Sloane - Sphere Packings, Lattices and Groups), so you just need the center density for similarity in this case.