Can’t add script immediately after the opening tag on login page?

Question

immediately after the opening <body> tag

WordPress doesn’t provide a hook for that, i.e. for adding code exactly after the <body> tag.

But WordPress does fire a hook named login_header which we can use to add code after the body tag is opened:

do_action( 'login_header' )
Fires in the login page header after
the body tag is opened.

So for example:

add_action( 'login_header', 'wpse_375598' );
function wpse_375598() {
    ?>
        Add your HTML here.
    <?php
}

And BTW, login_body_class is a filter hook one can use to add custom CSS classes into the <body> tag (i.e. <body class="here">), so that hook should not be used to add the (Google Tag Manager’s) noscript code.

Also in the edited question, the problem is that you used login_head and not login_header — login_head is for adding code in the <head> section of the document/page.

So your code should look more like:

add_action( 'login_head', 'wpse_375598_login_head' );
function wpse_375598_login_head() {
    ?>
        Add your HTML here that goes in the <head>.
    <?php
}

add_action( 'login_header', 'wpse_375598_login_header' );
function wpse_375598_login_header() {
    ?>
        Add your HTML here that goes in the <body>.
    <?php
}