Theming breadcrumb.html.twig
One quick option is to add some html markup that wraps the region (if you've disabled the default Drupal theme markup) in your page twig template where you call {{ drupal_block('breadcrumbs') }}.

For instance,

&lt;div class="topnav"&gt;
{{ drupal_block('breadcrumbs') }}
&lt;/div&gt;


and

&lt;div class="leftnav"&gt;
{{ drupal_block('breadcrumbs') }}
&lt;/div&gt;




Alternatively, if your theme is setup to render blocks assigned to regions in Drupal's Structure > Block Layout, you could:


install/enable the Block Class module (as of Feb 23, it's in an alpha release for D8) 
place two Breadcrumb blocks, one per region
assign your topnav and leftnav classes in the block config screen






For both use cases, you'll need to change your css to target:

.topnav ul
.topnav ul li
.leftnav ul
.leftnav ul li