in the same way: doing or being like what has just been described, so too, likewise
This is a small connecting word that points back to something just said and claims that the next thing matches it. Its plain force is "in the same way": whatever happened, or whatever should be done, applies again here, in the same manner. English most often carries it with "likewise" or "so too."
The word does the same job across very different settings. It ties parallel items together in a story (a second brother marries the widow, and a third, and so on down the line), it stacks up examples in an argument, and it lines up parallel instructions in a household ("in the same way, wives...", "in the same way, husbands..."). In each case the point is correspondence: this new case belongs with the one before it and is to be understood, or acted on, the same way.
Because it is a pure word of comparison, it carries no meaning of its own beyond "matching what came before"; its content is borrowed entirely from whatever it is being compared to. That is why it reads so smoothly in translation and rarely needs explaining.