Among most speakers of English, "with" needs something to go with it.
"Can I come with? Can you take me with?" and similar questions are heard from some speakers, especially in areas influenced by settlers from Scandinavia or Germany such as Wisconsin and Michigan. (There, the preposition is not orphaned but, at least in German, part of a separable verb such as "mitkommen, mitnehmen".)
The usage strikes many other speakers of English as wrong, though, which is proabbly why Duolingo doesn't accept it, and required "Can I come with you? Can you take me with you?"
Similarly with other regionalisms such as "You want I should clean you car?" or "It needs fixed" or "It done broke".
When i moved to Georgia, i got used to hearing these phrases from Georgians and Tennesseeans!
Rather than the translation into English - If I was to say "Kan du tage mig" would that sound really strange to a Danish person?
That sounds like a half-done sentence. The 'med' is important for the meaning here, being some kind of 'along (with you)'. "Kan du tage mig" on its own has a bit of the same indecent meaning as the English "taking me".