Double or single quotation marks in tags

First of all, I’m not sure why you specifically want to use double quotes here. According to the HTML standard, single quotes are also allowed. But from a perfectionist’s perspective, I can at least understand the question. 🙂

For the robots tag, the output in this format comes from the WordPress code itself. It’s defined here: https://github.com/WordPress/WordPress/blob/master/wp-includes/robots-template.php#L49

Of course, you can’t – and shouldn’t – modify this file. One solution, however, could be the sledgehammer approach, which I’m reluctant to use but consider a last resort for something like this:

function custom_replace_quotes( $buffer ) {
    // Here you can change the HTML content
    return str_replace( "'robots'", '"robots"', $buffer );
}

add_action( 'template_redirect', function() {
    ob_start();
    ob_start( 'custom_replace_quotes' );
});

This involves modifying the robots tag. The other tags would then need to be adjusted in a similar way since they are dynamically generated by the core.

If you’d like to see a change in the core regarding this, feel free to submit a ticket to the developers here: https://core.trac.wordpress.org/newticket