Can I add custom attributes while adding inline scripts?

Here’s one demo suggestion:

add_action( 'admin_enqueue_scripts', function()
{
   wp_enqueue_script(    'my-script', '/my-script.js', ['underscore', 'backbone'], '1.0' );
   wp_add_inline_script( 'my-script', 'alert("hello world");' );

   // Add our template
   if( function_exists( 'wpse_add_inline_tmpl' ) )
       wpse_add_inline_tmpl( 
           $handle="my-script", 
           $id     = 'my-tmpl', 
           $tmpl="<div class="section intro">{{{ data.section.intro }}}</div>" 
       );

} );

where we define our custom wpse_add_inline_tmpl() function as:

function wpse_add_inline_tmpl( $handle, $id, $tmpl )
{
    // Collect input data
    static $data = [];
    $data[$handle][$id] = $tmpl;

    // Append template for relevant script handle
    add_filter( 
        'script_loader_tag', 
        function( $tag, $hndl, $src ) use ( &$data, $handle, $id )
        {   
            // Nothing to do if no match
            if( ! isset( $data[$hndl][$id] ) )
                return $tag;

            // Script tag replacement aka wp_add_inline_script()
            if ( false !== stripos( $data[$hndl][$id], '</script>' ) )
                $data[$hndl][$id] = trim( preg_replace( 
                    '#<script[^>]*>(.*)</script>#is', 
                    '$1', 
                     $data[$hndl][$id] 
                ) );

            // Append template
            $tag .= sprintf(
                "<script type="text/template" id='%s'>\n%s\n</script>" . PHP_EOL,
                esc_attr( $id ),
                $data[$hndl][$id]
            );

            return $tag;
        }, 
   10, 3 );
}

Here we use the script_loader_tag filter to inject the template code and use both the handle and id to reference it. We also use the script tag replacement from the core wp_add_inline_script() function.

Hope you can test this further and modify to your needs. Writing this as a class might be a better approach 😉