They are already trading dolar token called tether that is backed 1 to 1 with real usd. They will not have a real usd, but they might have a real euro so it is fine. I am waiting for more fiat pairs, so we can finally break out from btc price.
What other people say about why Binance doesn't want fiat currencies in their work make sens, so I don't see reason why they would allow Euro and not Dollar... That would be just contradictory.
As for Dollar token, well that is just another crypto, so its fine if they accept it. But I doubt that altcoin prices will break from Bitcoin influence anytime soon... if ever.