It's because they want you to match the tense. "Vuoi" = "You want," not "you would like." "Would you like" would be "vorresti" or "vorreste" (I think, though I haven't actually learned that yet). We use them pretty much interchangeably in English, and I imagine they do in Italian as well, but there is a difference. I think the idea is that you know what tense you're actually using.
In English, "together" implies "with the speaker" So you could say:
Do you want to see the match together? (Though I'm American, so I would probably say "watch the game together.")
Do you want to watch the game with me/with us? (without "together")
To say "together with me" in this case sounds redundant to me.
I don't know in English but I agree with you that in Italy we would be more specific, saying
"... insieme a me" or
"... insieme a noi", or
"vuoi vedere la partita con me" or
"vuoi vedere la partita con noi".
Anyhow you can say even "insieme" and nothing more.
Or even in this way: "Vuoi vedere insieme la partita ?"
I all these ways is correct, and probably duolingo wanted to make things more easy.
"Would you like" is polite, and "Do you want" is not very polite at all, but it depends on how you say it (tone of voice).
You could say, "What do you want? Red or blue?" when you're annoyed or angry and it would come across as rude, impatient, demanding, etc. However, you could say the same thing in a calm and normal voice and it wouldn't be rude.
I would only use "Do you want" with family and friends. I would not use it with strangers or people you tend to be formal/polite with.