The Hausdorff dimension satisfies $$ \dim A + \dim B \leq \dim (A\times B) $$ while the (upper) packing dimension has $$ \dim_p (A\times B) \leq \dim_p A + \dim_p B $$ These inequalities hold for any subsets of the Euclidean space.

Do we know any fractional dimensions with the equation instead of the inequalities?

Any arguments where it would help to have such a dimension?

For 1. all I can find are statements of the kind: if $A$ and/or $B$ are good enough then one or both of the inequalities above become equations, but nothing for general subsets.

For 2. I found a paper by Darji and Keleti "Covering $\mathbb{R}$ with translates of a compact set" Proc. Amer. Math. Soc. 131 (2003), no. 8, 2598–2596.

It seems I might be getting such a dimension, but I can't find any context for it.