It's actually easier to go the other way around. Finite dimensional modules over an algebra $A$ fulfils the Krull-Remak-Schmidt theorem of being isomorphic to a direct sum of indecomposable modules with the indecomposable factors unique up to isomorphism. If now $L\bigotimes_KU$ and $L\bigotimes_KV$ are isomorphic as $L\bigotimes_KA$-modules they are also isomorphic as $A$-modules but as $A$-modules they are isomorpic to $U^n$ resp.\ $V^n$ where $n=[L:K]$ and by the KRS-theorem this implies that $U$ and $V$ are isomorphic. The generalised Thm 90 now follows as the cohomology set classifies $B$-modules whose scalar extension to $L$ are free of rank $1$.

It's actually easier to go the other way around. Finite dimensional modules over an algebra $A$ fulfils the Krull-Remak-Schmidt theorem of being isomorphic to a direct sum of indecomposable modules with the indecomposable factors unique up to isomorphism. If now $L\bigotimes_KU$ and $L\bigotimes_KV$ are isomorphic as $L\bigotimes_KA$-modules they are also isomorphic as $A$-modules but as $A$-modules they are isomorpic to $U^n$ resp. $V^n$ where $n=[L:K]$ and by the KRS-theorem this implies that $U$ and $V$ are isomorphic. The generalised Thm 90 now follows as the cohomology set classifies $B$-modules whose scalar extension to $L$ are free of rank $1$.