Equivalence of states between two "quasi-deterministic" strongly connected Büchi automata accepting the same $\omega$-language
Your property does not hold.

Consider the following languages over $\Sigma = \{a,b\}$:


There are infinitely many $a$s at even positions in a word
There are infinitely many $a$s at odd positions in a word


Both are representation as DBAs with the same set of states (of size 3). We can build DBAs for them that only differ in their initial states.

If we want to build an automaton for the disjunction of these languages, we can do so by taking both initial states for both of them as new initial states. This gives us a qDBA $B_1$.

The disjuction is equivalent to the language "there are infinitely many $a$s", for which a DFA $B_2$ only needs two states, each of which have the same language.

Now, $B_1$ and $B_2$ are a counter-example to your conjecture.