Answer: During the 9 days itself we do not use wine. Shulchan Aruch OC 551:10 quotes a Minhag that even though we do not eat meat during the 9 days; however, a food cooked with meat may be eaten without the meat, for example cholent potatoes. The Mishna Berura s.k. 63 differentiates that to add schmaltz would still be forbidden. In any event, he concludes, the Minhag today is not to eat any food cooked with meat itself, unlike the Shulchan Aruch.
Presumably, adding actual wine to a soup would be more similar to adding schmaltz than to a tavshil basar with removing the meat. However, the Be'er Hetiv 29 quotes the Taz who permits adding wine according to the Shulchan Aruch's Minhag. The Be'er Hetiv concludes that according to our more stringent Minhag this would also be forbidden. Apparently, he compares adding wine to a recipe to a tavshil basar and not to schmaltz.
Motzei Tisha b'Av we do not eat meat or drink wine, similar to the entire 9 days. However, the Biur Halacha in the beginning of 558 quotes the Maamar Mordechai to permit a tavshil basar, and seemingly agrees that to be lenient. Though one could argue that the Maamar Mordechei is following the Minhag of Shulchan Aruch to permit this the entire 9 days, the Biur Halacha does not seem to accept this. Since the Be'er Hetiv was only machmir not to rely on the Taz's permitting wine due to the Minhag and the Minhag is different Motzei Tisha b'Av than the rest of the 9 days, adding wine to the soup should be allowed.
Certainly, is the wine is less than 1/60th of the soup, or even if it is more but does not add a distinct taste of wine to the soup, it is unquestionably allowed.