Semantic equivalence using a model of computation of two languages
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:


$trans_A : A \rightarrow T$.  
$trans_B : B \rightarrow T$.


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


For each $A$-program $P_A$ there is a $B$-program $P_B$ such that $trans_A(P_A) \cong_T trans_B(P_B)$.
Vice versa.


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).