Let me clarify the setting, which has nothing to do with $\pi$-calculus or bisimulation.
The first thing you have to realise that it does not make much sense to talk about a programming language without reference to the notion of program equivalence you intend to impose on the language. That's because
We usually identify certain programs (e.g. f(x:int) = { x+1 } is considered the same as f(y:int) = { y+1 }).
The notion of program equivalence to be used depends highly on what you are doing with these languages, e.g. sometimes we care about program time- or space complexity, sometimes we don't.
So the general setup you are interested in is this: you are given 3 language $(A, \cong_A)$, $(B, \cong_B)$ and $(T, \cong_T)$, each comprising of a set of programs ($A$, $B$ and $T$) with associated notions of program equivalence ($\cong_A$, ...). In addition, you have two mappings:
They map ("compile") from source languages $A$ and $B$ to target $T$. Typically you want these two translation functions to be at least sound. That means that for all $A$-programs $P, Q$, we have: $trans_A(P) \cong_T trans_A(Q) \Rightarrow P \cong_A Q$, and likewise for $trans_B$.
Now you want to argue that $(A, \cong_A)$ and $(B, \cong_B)$ are equivalent as languages. That's what you use the target $(T, \cong_T)$ for. There are many ways that this can be done, details depend on what concept of language equivalence you are interested in. One approach could be to say
None of the above is $\pi$-calculus specific, but you
can instantiate $T$ with $\pi$-calculus and $\cong_T$ with any chosen notion of equivalence on $\pi$-calculus such as strong or weak bisimulation, maximum consistent theory, reduction congruence, or whatever else makes sense for your application.
I said that there are many alternative ways of doing this. You could for example impose constraints on $trans_A$ and $trans_B$ so the translation preserves chosen constraints (e.g. termination, or compositionality).
