Firstly, the sessionStorage already stores a string, and the var you are saving into, shouldn't be called sessionStorage, because it hides the name of the original sessionStorage. Also avoid using "var" alltogether, use const and let.

const shoppingCart = sessionStorage.getItem('shoppingCart')

Secondly, the data you are sending is URL encoded into the body of the HTTP request (and the JSON part probably gets scewed up).

encodeURIComponent(shoppingCart)

My recommendation would be to encode everything in JSON. Since you are already "lying to the protocol" with the dataType: "json" you provided (your actual data is application/x-www-form-urlencoded not application/json).

JS:

const shoppingCart = JSON.parse(sessionStorage.getItem('shoppingCart'))

$.ajax({
  url: "https://mywebsite.com/wp-admin/admin-ajax.php",
  method: "POST",
  data: JSON.stringify({
    globalvar: shoppingCart,
    action: "myfunction"
  }),
  dataType: "json"  
}).done((res) => {});

PHP (🤢):

$json = file_get_contents('php://input');
$body = json_decode($json);

$func = $body->action;
$data = $body->globalvar;

To read application/json in php we can use php://input

So the code will be like this

function myfunction() {
        $rawData = file_get_contents("php://input");
        $json = json_decode($rawData);

        $object = $json->globalvar;

To send application/json from browse to php we can use JSON.stringify

So the code will be like this

sessionStorage.setItem("shoppingCart", JSON.stringify([{"name": "name1"},{"name": "name2"}]));

var dataStorage = JSON.parse(sessionStorage.getItem('shoppingCart'));

$.ajax({
    url: "test1.php",
    method: "POST",
    data: JSON.stringify({ "globalvar": dataStorage, "action" : "myfunction"}),
    dataType: "json"  
}).done(function (response) {
    console.log(response)
});

There are several aspects to fix/check:

1. Is the function defined as you want?

Your

if (!function_exists('myfunction')) {
    //...
}

Checks whether myfunction exists and only executes he inner block if a function by this name does not already exist. You will need to make sure that your function ends up existing and some older function by the same name does not prevent this function from being created.

2. Is your JSON in proper format?

You are json_decode'ing your session data and on the database level you would need to store a value like

'[{"name":"том","price":259,"count":1},{"name":"жак","price":299,"count":1}]'

into your record. Try to make sure that the way you are storing your JSON is valid in Wordpress.

3. Check the Network tab of your Dev Tools

You will need to make sure that your request is being sent properly and the JSON you expect to be sent is being sent after all.

4. Make sure myfunction is being executed after all

It would not hurt to add some logging to this function to see that it's being executed.

I think sessionStorage is a reserved keyword so it is creating the issue. You can debug the variable state. This code works for me.

var dataStorage = JSON.stringify(sessionStorage.getItem('shoppingCart'));

$.post({
  url: "https://mywebsite.com/wp-admin/admin-ajax.php",
  method: "POST",
  data: 'globalvar=' + encodeURIComponent(dataStorage) + '&action=myfunction',
  dataType: "json"
}).done(function(response) {});

PHP Code - Just removed a few lines for debugging

if (!function_exists('myfunction')) {

    function myfunction()
    {
        $object = $_POST['globalvar'];
        $decoded_object = json_decode($object);
        print_r($decoded_object);
    }

    myfunction();
}