Hey there,
The reason that it doesn't store your template override automatically into esmobile template is because esmobile is not set it as frontend default template.
That 'Override File Location' in the 'Editing Email Template' only saves for your current frontend default template only.
Actually you can ignore this, as long as you have done for the template override from your Vanilla template, you can just open and copy this directory folder JoomlaFolder/templates/vanilla/html/com_easysocial, then paste into your mobile web template directory folder JoomlaFolder/templates/esmobile/html/com_easysocial.
Can you have a try and see how it goes?