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I am almost sure that the answer is no. Your question is strongly related to Keller's conjecture which turned out to be false. There a tiling of $\mathbb Z^{10}$ was given by translates of the unit cube such that the center of each translate is a halfinteger. Moreover, the tiling has a very strong lattice-like structure, 1024 cubes whose center is in $[0,2]^{10}$ are translated by every vector from $2\mathbb Z^{10}$.

Of course, the cube won't give a counterexample to your question. But you can divide each cube from this construction to like $3^{99}$ little cubes and then "dig little tunnels" among adjacent cubes to make sure the resulting polyominos only fit in the given way. This leaves many details to work out and I am not 100% sure it can be done but looks like a promising approach.
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I am almost sure that the answer is no. Your question is strongly related to Keller's conjecture which turned out to be false. There a tiling of $\mathbb Z^{10}$ was given by translates of the unit cube such that the center of each translate is a halfinteger. Moreover, the tiling has a very strong lattice-like structure, 1024 cubes whose center is in $[0,2]^{10}$ are translated by every vector from $2\mathbb Z^{10}$.

Of course, the cube won't give a counterexample to your question. But you can divide each cube from this construction to like $3^{99}$ little cubes and then "dig little tunnels" among adjacent cubes to make sure the resulting polyominos only fit in the given way. This leaves many details to work out and I am not 100% sure it can be done but looks like a promising approach.