A few more:
DISTRICT VI
Bortodoor https://www.bortodoor.com/ Also pretty casual. Run by an American, UK and Aussie. Wednesday is blind tasting night. Menu is brought in from local restaurants.
Cork Breakfast and Wine https://www.instagram.com/cork_breakfast_and_wine?igshid=YmMyMTA2M2Y%3D Excellent menu and wine selection. Ukrainian owned and run. Next to the Opera House.
Champion Wine https://www.championwines.store/about/ “Champion Wine is shop, cellar and gallery in the heart of Budapest, it offers a selection of over 460 Hungarian wines. Hungarian wines with ham and cheese plates.
Kadarka Wine Bar and Wine Shop (Kadarka Borbár és Borkereskedés) http://kadarkawinebar.com/ Mostly Hungarian wines made from long-forgotten Hungarian grape varieties, both for tasting and takeaway on site. It’s Casual and attracts young to old. You can’t beat the selection, the service or the prices. The waiters know their wine well and this is the one where you can sample a few before you buy glass. Food menu is very good as well.
Marlou Wine Bar and Store https://www.marlouwinebar.com/ Natural Wines. They promote natural wines: those that express their place of origin (wherever they come from), hand-crafted wines with minimal or no intervention and in which the winemaker shows maximum respect.
DISTRICT VII
Dobolo https://www.facebook.com/doblowinebar Nice upscale wine bar. “Doblo Wine & Bar is specializes in wine tastings with fingerfood /delicious, hungarian hand-crafted cheese and meat with pickles and bread/. Every day live music. Mainly Jazz, Funk, Gypsy jazz and pop duos or trios playing some excellent background music from 8.30pm.
DISTRICT XI
Palack Borbár https://palackborbar.hu/ They stock 80-100 wines, usually from smaller wineries. Good tapas and a sweet view of the Gellert Hotel, the Danube and the Szabadság bridge. A nice escape from the tourists.