You mean this?

function howdy_doody(person) {
  let compareValue = 'dog'
  if (compareValue == person.name) {
    return 'hello'
  } else {
    return 'goodbye'
  }
}

howdy_doody({ name: 'dog' });

function howdy_doody(person) {
  let person = 'dog' // <-- this line overwrites the input `person`; remove it
  if (person == { name: 'dog' }) { // <-- if (person.name === 'dog') {
    return 'hello'
  } else {
    return 'goodbye'
  }
}
howdy_doody({ name: 'dog' });  // <-- "hello" gets returned here, 
                               // but you're not assigning it to
                               // anything, or logging it, so there
                               // won't be a visible result

(...but above I'm assuming you actually mean to check that person has a name param with value 'dog'; if you wanted to check that person is exactly equal to the object {name:'dog'} that gets a little more complicated.)

For an exact match of { name: 'dog' } one could e.g. utilize Object.entries which for any given object returns an array of this object's own key-value pairs / entries. Each entry gets provided as a key-value tuple.

Thus the entries array length value should be exyctly 1 and the key of this sole tuple has to equal 'name' whereas the value has to equal 'dog'.

function isValidHowdyType(value) {
  const entries = Object.entries(value ?? {});
  return (
    entries.length === 1 &&
    entries[0][0] === 'name' &&
    entries[0][1] === 'dog'
  );
}
function howdy_doody(value) {
  let phrase = 'goodbye';

  if (isValidHowdyType(value)) {
    phrase = 'hello';
  }
  return phrase;
}

console.log(
  howdy_doody({ name: 'dog' })
);
console.log(
  howdy_doody({ name: 'dog', owner: 'Jane' })
);
console.log(
  howdy_doody({ name: 'cat' })
);
console.log(
  howdy_doody({})
);
console.log(
  howdy_doody(null)
);
console.log(
  howdy_doody()
);

.as-console-wrapper { min-height: 100%!important; top: 0; }