The description text there is not clearly written, but I think the comment means that entering the synonyms and completing a lesson must be done on the same PC because the synonyms will only be uploaded after you completed the quiz. So if you enter the synonyms on PC and later complete the quiz somewhere else it will not work. But synonyms that are successfully committed with the script will show up on other devices or kaniwani.
Edit: there is some discussion here, you can only retrieve the synonyms from the server after you finish the quiz, “only in one browser” only refers to seeing your new synonyms after you added them and before you finished the quiz. They are uploaded immediately 