php – How to save custom Session variable till the checkout page?

I am trying to save a session variable based on a query parameter.
Example – https://ferroglan.com/cover-search-result/?swoof=1&maker=a3

I want to save maker=value throughout the single session until the browser is closed or the search parameter changes.

What I have achieved so far is that on the result page I can see the value of maker in my variable.
session variable

But as soon as I move to the product page it doesn’t stay even though code is present on the head tag.

Q. How do I save it on the product page as a custom variable and then till the checkout so that when someone places the order I can know which model the user has selected in the Admin dashboard.

// Get query parameter 

  add_action('init', 'session_maker');
  function session_maker()
  {
          session_start();
    if(isset($_GET('maker')))
    {   
        $_SESSION('maker') = $_GET('maker');
        echo $_SESSION('maker');
    }
    
  } 

I know to add the custom field to the woo-commerce checkout but mainly I want to understand how can I save this variable value to the product page.