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True. The valid distributive law for ordinal multiplication isThe synthetic definition of ordinal multiplication takes a sequence of copies of the left factor, indexed by the right factor. A sequence indexed first by $\beta$ and then by $\gamma$ is exactly the concatenation of the two corresponding sequences. Taking order types gives the identity. Equivalently, use transfinite induction on $\gamma$, with the successor rule and the supremum rule at limit ordinals.

False. Put $\alpha =\beta =1$ and $\gamma =\omega$. ThenThe two ordinals differ. The order of the factors matters in the distributive law for ordinal multiplication.