My website operates in two languages. I intend to provide language-specific subscription forms for users. How do I create multiple subscription forms and connect the forms to individual list?
you can create custom forms (read our documentation on how to do that) and link them to different lists, then place them in the translated pages according to your site languages settings.
Next, I’m curious about the “inside posts” and “popup” forms. I’ve enabled them, but they seem connected to the “Standard” tab form. Can I assign the specific language form(shortcode) I built for these features?
yes, those are connected to the standard form only. Anyway, if you created custom forms you can use the generated html pretty much everywhere, i’d recommend trying some popup/injection third party plugin, which will do the trick.