"The Night Sisters" is a plural noun. Therefore the possessive form is "The Night Sisters'", adding just an apostrophe. The proper form of thre example sentence is
Mary tended John throughout the Night Sisters' passing.
Similarly with other plural nouns:
I greatly enjoyed the Bermuda Islands' sunny beaches.
"the Bermuda Islands" is a plural noun
Pages run the Senators' errands.
"Senators" is a plural noun.
Note that this rule applies to all plural nouns. There is not a special rule for twins, but twins will often be referred to in the plural.
However, a singular noun that happens to end in the letter s does not follow this rule (Strunk and White, The Elements of Style, Rule 1: Form the possessive singular of nouns with 's.")
- I enjoyed reading Charles's books.
- The Shores's menu is quite extensive.
- The actress's performance drew loud applause.
- Chess's rules are more complex than those of the game Go.
Many native speakers ignore this rule, but in my view this is a mistake.
Often the possessive form can be avoided by rewriting. For example "The rules of chess" rather than "Chess's rules". Sometimes this improves the flow of the sentence.
To clarify the title question: A "twin" is a single person who is one of a pair or set of twins. The term for a set is 'twins" or "a pair of twins", which is plural, and treated as such. ("twins" can also mean a larger grouping of people who are twins, either in sets or not.) Thus
The twin's clothing
means the clothing of one person, who happens to be a twin, while
The twins' clothing
mean the clothing associated with a set of twins, or possibly with several sets. Given how easily these forms can be confused, if the distinction is important, the sentence should be rewritten for clarity, probably by avoiding the direct possessive. Thus:
Fred and John are twins. Fred's clothing is mostly red and white, while John's is blue and white.
Jane and Sally are twins. Their clothing is largely green and yellow.