" So is " means " also " and " too" [a positive statement] When we use " So is" to make short agreements, it is followed by an inversion. Sentence Structure : So + BE (is, am, are, was, were etc.) + Subject { Present Simple Tense : is - Singular Subject / am - I / are : Plural Subject } { Past Simple Tense : was - Singular Subject / were - Plural Subject } e.g . John : I am fond of Techno Dancing. Jane : So is Peter. ( Peter = Singular Subject) " So is Peter " comes from { " Peter is also fond of Techno Dancing" } so we use Inversion to make the sentence shorter [concise] and avoid redundancy in order to add more information . We can also use this inversion with other auxiliary verbs Sentence Structure : So + auxiliary (do, does, did, will etc.) + Subject e.g. After the cinema :- Jane : I enjoyed the movie. John : So did I.

So is = also. In your example, the speaker is saying that their disposition is also sometimes foggy like the weather. Here are some more examples~ A: "Ben's tired." B: "So is Aaron." A: "The kids are hungry." B: "So are the parents."