A currency's strength or weakness is only meaningful in terms of confidence in its future buying power. And that is a reflection of a country's monetary policy, not of the specific units used to quotes prices.
Not exactly regarding the dollar. If oil trading turns to Euros that reduces the demand for dollars. Through inflation, the dollar is worth less to foreign countries who have dollar reserves. These countries basically get stuck with a tax with inflation and want to have less dollars. With a drop in demand worldwide for the dollar, there so goes the worth of the dollar.
Why? (Beyond transactions costs associated with using exchange rate markets.)
It is one thing to say, "people don't have confidence in the dollar, and therefore an exchange in euros will atract more users;" and another thing to say, "because a anti-US regime insists on (subsidizes?) an exchange that officially quotes prices in euros, therefore people will lose confidence in the dollar."
Anyway, rather than strong versus weak, I think the better way of classifying currencies is hard versus soft. Iran can subsidize every barrel exchanged on its bourse and we'd see large volumes trade, but it would not mean much at all. (The ease-of-transactions premium on dollars is not zero, but I would have to see some evidence that it is econometrically detectable.)