1 It looks like rain. -- No, I don't think so (I don't think that it is so/in such a way).
2 We are doctors. -- So are we. (We are doctors as well. Compare German ebenso: One might say in German: Ebenso sind wir Ärzte. It is not idiomatic today, but everybody would understand it. Today one would say: Wir auch.)
The use of "so" in "So are we" for "We too/We as well" is a bit curious. But it is a common manner of answer. It is a bit cumbersome to reconstruct a longer formula with "so". One would have to make two sentences: So/in the same way it is true for us. We are doctors too.
Please take all this as a hypothesis of mine. An attempt to make such uses of "so" better understandable and to show that we actually have the normal use of "so" meaning "in such a way".