Remove all styling from views field view?
Yes, you have to override the templates of the child view to remove the markup you don't want.

You will might need to override (in order of how likely it is you need to override them in some way):

views-view.tpl.php
views-view-unformatted.tpl.php (if you are using the unformatted style. Otherwise whichever style template is relevant to your view)
views-view-fields.tpl.php

You need to find where the unwanted line is coming from in the markup/css and then find the template that markup is printed in, then override that template and remove the offending markup.

Alternatively you can likely style it out with css but I would favour the template override option.