Both could be used, but the plural sounds more natural to me because it's an invitation to the other person for the two of you to be partners. I'd like (for us) to be language partners.

I think they're both correct - usage may differ between speakers of different varieties a little. Saying 'partners' means he's suggesting that you'll become language partners together with each other, whereas 'partner' (which is how I would tend to phrase it) emphasises that s/he would like to be your language partner. The meaning is the same but with a subtle change in emphasis (which it's likely the speaker wouldn't consciously make).

The sentence that was written is wrong. I would like to be your language partner. Once the request is accepted, you are language partners.