Which wordpress function is the caption shortcode linked to?

You should try this on a default theme (without any plugins), to see if that makes any difference.

Your shortcode example:

    <img src="https://wordpress.stackexchange.com/questions/158628/whatever.jpg" alt="whatever" width="399" height="600" /> 
    My caption text

gives this output:

<div id="my-caption-id" 
     style="width: 409px" 
     class="wp-caption alignnone my-caption-class">
    <img src="https://wordpress.stackexchange.com/questions/158628/whatever.jpg" alt="whatever" width="399" height="600">
    <p class="wp-caption-text">My caption text</p>
</div>

on the Twenty Twelve theme. It’s just a little bit formatted to make it easier to read here.

So the caption id and classes are displayed in the output.

Update:

It looks like your theme is overriding the image caption via:

// Remove default inline style of wp-caption
add_shortcode('wp_caption', 'fixed_img_caption_shortcode');
add_shortcode('caption', 'fixed_img_caption_shortcode');

in the /library/cleanup.php file.