jquery Use two done callbacks in same function, one with datatype json one without

Yes, you can nest ajax callbacks. But first we may want to review what we're talking about here.

JSON stands for javascript object notation. It isn't a data type, it's a way to encode a javascript object into a text string so it can be easily stored/transferred. https://www.w3schools.com/js/js_json_intro.asp

jQuery is a javascript library. It's basically sytactical sugar for vanilla javascript. https://jquery.com/

PHP is a sever side language used for taking client requests, doing stuff with it, and returning a response. You don't call php functions from the client. You make a request and the sever decides how to respond. https://www.php.net/

That out of the way, if you want different data back from the same url, you would have to add that in your options object and then handle that server side. Client side would be something like this:

jQuery.post(my_ajax.ajax_url, { action: 'myphp', return_type: 'string' })
  .done(function(data) { //data will be a string
    jQuery('#adivID').html(data);

    //this part doesn't make sense, because if the data is a string,
    //it won't have an 'id' property. But maybe in your actual code this
    //does make sense. So this is just an example of how nesting works.
    jQuery.getJSON(my_ajax.ajax_url, { ids: data.ids, return_type: 'json' })
      .done(function(json) { //json will be an object
        //do stuff with the object
      })
      .fail(function() {
        //response was probably not properly json formatted
      });
  })
  .fail(function(xhr, status, error) {
    console.log(xhr.responseText);
    alert(error);
  });

jQuery.post() and jQuery.getJSON() are just short hands for jQuery.ajax(), as they require less parameters, and unless you're doing something more complex, keeps your code more readale.

Edit (since you added your php code):

I'm not familiar with wordpress, and it took a few reads to try and figure out what your code is doing. But what I gather from the docs, you might want something like this:

$ids = $wpdb->get_results("SELECT id FROM mycats", ARRAY_N); //returns an indexed array of ids

switch($_POST["return_type"]) {
  case "string": //returns a comma separated string of ids
    echo implode(", ", $ids);
    break;
  case "json": //returns a json array of ids
    echo json_encode($ids);
    break;
}

But again, the getJSON() is going to fail because the post() is going to return a string. It's tough to suggest code without knowing exactly what it is you're trying to accomplish. It's probably also worth noting that in php an object is different from an associative array is different from an indexed array. Also, all variables start with a $. wpdb != $wpdb If you haven't lost several hours because of this, you haven't written enough php haha.