Everett’s proof that the Born rule measure (amplitude squared for orthogonal states) is the only measure that satisfies the desired properties has no dependence on tensor product structure.
Everett’s proof that a “typical” observer sees measurements that agree with the Born rule in the long term uses the tensor product structure and the result of the previous proof.
I will amend my statement to be more precise:
Everett’s proof that the Born rule measure (amplitude squared for orthogonal states) is the only measure that satisfies the desired properties has no dependence on tensor product structure.
Everett’s proof that a “typical” observer sees measurements that agree with the Born rule in the long term uses the tensor product structure and the result of the previous proof.