'90s is a contraction of 1990s, or whatever other century's tenth decade is implied. So the apostrophe belongs as an example of a contraction or possessive :)

But your apostrophe is upside down (it became a curved single-open-quote in your version). It’s supposed to be this way:

’90s

Just 90s if it's a reference to someone's age. And the apostrophe for referencing the 90s era is optional outside of formal writing and a dash can also be used e.g.; 90s-music depending on context.

As bitjunkie said, if you were referring to 90s in the possessive, it would be 90s' since it's a plural noun already - much like parents' mortgages or stores' buildings.

That said, I would probably look at the phrase "90s child" as either (1) a compound noun not needing anything to be possessive (like "ice cream"), or (2) with "90s" as an adjective modifying "child" (like "latchkey kid").

No because it's plural so the apostrophe would come after the S