This thread is resolved. Here is a description of the problem and solution.
Problem: When I go to the UI of the theme, the English version shows branches in both languages, and the same goes to Arabic, it shows English also.
Solution: According to this article from the plugin documentation https://welaunch.io/plugins/wordpress-store-locator/docs/faq/wpml-support/, it seems that the way that the plugin works (and gets the list of branches) is not compatible with WPML. Some changes inside the plugin need to be done in order to retrieve the list of branches independently on each language.
This is the technical support forum for WPML - the multilingual WordPress plugin.