You're looking for a Set intersection -
function intersection2(a, b) {
  if (a.size > b.size)
    return intersection2(b, a)
  const r = new Set
  for (const v of a)
    if (b.has(v))
      r.add(v)
  return r
}
You can compute the intersection of many sets by combining reduce and intersection2 -
function intersection(s, ...sets) {
  return sets.reduce(intersection2, s)
}
To use it with your code, get all values of x using Object.values and map over them to create an array of sets. Finally call intersection with all of the sets -
const x = {
  c: ['Full_Name', 'Last_Name', 'Email', 'Organizations', 'MISC', 'UNIQUEID'],
  small: ['Title', 'Organizations', 'URL', 'UNIQUEID'],
  big: ['abc', 'Organizations', 'def', 'UNIQUEID']
}
const result =
  intersection(...Object.values(x).map(v => new Set(v)))
console.log(result)
Set [ "Organizations", "UNIQUEID" ]
If you want the result as an array, you can convert it using Array.from -
console.log(Array.from(result))
[ "Organizations", "UNIQUEID" ]