Align embedded Media Images
The easiest to resolve (and most likely) issue is:

You are limiting the allowed HTML tags for the text format you're using and you haven't allowed the align (and also caption, if you want captions) attributes for the drupal-entity HTML element. (I'm assuming you do have the Align images and Caption images filters enabled because you said those features were working for non-media embeds. If they're not enabled, enable them.)

Solution:

In the "Limit allowed HTML tags and correct faulty HTML" section (/admin/config/content/formats/manage/MY_FORMAT#edit-filters-filter-html-settings), make sure the drupal-entity has at least the data-align (and data-caption for captions) attributes. E.g. &lt;drupal-entity data-align data-caption data-etc...&gt;.