I also have a hard time understand this: why?! It is if the German course would teach dated the dative -e ending for masculine and neuter nouns (zum Tage, auf dem Tische) or if the Swedish course would teaching the dated way of conjugating verbs in plural (vi äro, vi gingo till skogs etc.).
According to Google Ngram Viewer, the usage of со мною has dropped ever since the beginning of the 20th century and is still dropping in favour of со мной.
With all this being said, I actually like being ‘old-school’, using slightly dated terms and expressions, just as ‘homewards’ in English or ‘den gamle mannen’ in Swedish.
I skipped the bulk of the beginning of this course, so it's possible I missed something, but unless there's a lesson on slang or verse, I would remove this from the course. Pretty much the only time you will hear this in my opinion is in songs, when they need to make something rhyme or meet a particular meter.