No, it would be Jesteśmy tacy samotni.
Sam/sami == alone or same, but you use them completely different than in English. 'I am tired' ' Same' => 'Jestem zmęczony' 'Ja tak samo'.
This is one of those words that is hard to explain to foreigner, but natives understand different meanings and know when to use it.
The translation was pretty much spot on. If we were literal, it would be "I the same way".
The word "tak" in the example you provided is used to signify a method. Somebody might answer the "jak" (how) question with "tak" while showing or demonstrating the method. I think it's slightly different from "tak" in the "so much" sense.