A possessive form (more formally a "saxon genitive") can mean actual ownership, or some kind of relationship.
- Children's upbringing = the upbringing that children receive
- An election's results = the outcome of the election
- Down's Syndrome = the condition that Down fist described, now often called "Down Syndrome to avoid the possessive.
- Darwin's finches = a group of birds that Charles Darwin first formally described
- David's book = the book that David owns
- Stephan Jay Gould's books = the books that Gould wrote
- The Earth's orbit = the path that the Earth follows through space
Whether a "possessive" form is most commonly used, or an adjectival form such as "election results" is used, or a form such as "results of the election" is largely a matter of historical accident. grammatically these have the same meaning. it is just a matter of which form became popular and therefore common, which may be influenced by the source of the phrase.
(Note that "children upbringing" is a common form, but one says "child rearing" not "child's rearing" or "a child's rearing".)
By the way one should not write (as the OP originally did before the question was edited) "No one write election's results"; rather one should write "No one writes 'election's results'" because "no one" is singular.
Quoted or referenced words should ideally be marked off with quote marks, italics, block quotes or other methods. This is a form of the use-mention distinction.