Hi,
Nothing changed in the emails, it's the address format which does not follow you when you change template.
HikaShop stores that format as a file inside the template folder :
templates/YOUR_TEMPLATE/html/com_hikashop/address/address_template.php
When you edited the address format in the configuration, that file was written into the templates installed at that moment. Vessel was not one of them, so now that Vessel is your default template, HikaShop finds no file there and falls back to its own default format, which does not include the complement line. Same thing in the checkout, since the address block there is displayed with that same format.
Only the display is affected. The field itself, its custom field settings and the data stored on the orders are untouched, which is why you still see the line when editing the address.
To fix it, go in the HikaShop configuration, Checkout tab, and look for the "Address format" area. Add your {address_street2} line back where you want it and save. Saving writes the file in every template installed on your site, frontend and backend, so Vessel will get it and so will the next template you install.
If you prefer not to retype it, the file is still in your previous template. Copy it to templates/vessel/html/com_hikashop/address/address_template.php and it comes back as it was.
One remark on the list of tags you posted : {address_line2} is not a HikaShop field name, unless you created a custom field with that column name yourself. The standard complement field is address_street2. Unknown tags are simply removed when the address is displayed, so it does no harm, but it does nothing either.