The Schoenberg criterion (en.wikipedia.org/wiki/Euclidean_distance_matrix) is relevant. It's a criterion for which a set of $n$ points in $\mathbb R^k$ can realize a given set of pairwise distances.

Fundamental to packing questions is balls in the metric we are interested in. Unfortunately it seems we are not even close to understanding the volume of ball in tensor rank metric.

Hmm, fair enough. Actually my motivation comes from coding theory. Classic coding theory is (roughly) about packing vectors over $\mathbb F_q$ in Hamming metric or packing vectors over $\mathbb R$ in Euclidean metric. It turns out similar packing questions can be posed for matrices over $\mathbb F_q$ in rank metric $d(\mathbf{A},\mathbf{B})=\mathrm{rk}(\mathbf A-\mathbf B)$. Such codes find applications in network coding. I think a natural generalization would be packing tensors in tensor rank metric. I am not aware of any application but maybe it is of independent interest.

However, I do not care about complexity. What I want is some formula for this number which is ``reasonably" easy to use. For instance, counting independent set is #P-complete, but the number of independent sets is given by a nice polynomial. Evaluating that is certainly hard but sometimes that is good enough for some mathematical reasoning.

Thank you Joshua. Indeed the hardness results do not appear surprising to me. For instance the title of this paper sort of says everything. I also agree with your algebraic interpretation. GI can be formulated as checking whether two matrices are in the same orbit which is no harder than the tensor version.

@NathanielJohnston Thank you Nathaniel. I think you are right. en.wikipedia.org/wiki/Tensor_rank_decomposition#Maximum_rank -- unlike matrix rank which is obviously bounded by the dimension of the matrix, the maximum rank of a tensor is unknown and we do not even have a conjecture for that. However, according to wiki, this is the situation over $F\in\{\mathbb R,\mathbb C\}$. Since the tensor rank heavily relies on the underlying field, do you know whether anything more is known over $\mathbb F_q$? Even knowing a tensor rank bound would be somewhat good to me.