Short quiz about Myhill-Nerode theorem
You are mistaken. Strings $x$ and $y$ are equivalent modulo the language $L$ iff for  all suffixes added to the words the results are both in or both outside of the language:

$x\sim_L y$ iff $xw\in L \iff yw\in L$ for every $w$.

In particular, when the strings $x$ and $y$ end in the same state for the (deterministic) automaton $M$, then $x\sim_{L(M)} y$, as they can be extended by exactly the same strings. This also holds for the dead-end state where no strings can extend the original string to acceptance.