As the title says,
There is a specific Form Container that I would like to add a twig to that is on a page having its own twig. No theme hook name suggestion is provided when I inspect its element.
My purpose is to style the specific form to be a credit card form.
You could try to add your own template suggestion via hook_theme_suggestions_HOOK_alter, which would mean hook_theme_suggestions_container_alter. There you might target the container by id or some other attribute (id does not seem promising here).
The container.html.twig does only provide a wrapping div around other content. While you could add more html to this, you might have to think about if container.html.twig is the right place to start here.