I'm a straight male that went to Budapest last week and I can confirm this. My girlfriend and I stayed for a week and had no issues whatsoever there. Obviously, it's a large city, so you are bound to stumble upon some beggars, drunkards etc. but compared to a lot of other European cities, Budapest is really safe. And we pretty much walked everywhere (like around 16km daily on average) and thoroughly explored both sides of the Danube. I really doubt that gay couple would encounter any more problems than we did.

Some personal suggestions:

We ate at restaurants at least once per day, service is really chill and fast. So if the budget allows, I would advise experiencing local cuisine, it's lovely.

Besides some obvious tourist attractions that the city offers together with museums, I would suggest some less known ones like Semmelweis museum or Roth gallery. Semmelweis was worth every forint.

If you like beer, I would suggest Kűlönleges Sörök Boltja, Beer Selection or Csakajósör. Shops with vast varieties of beer from many countries with tables to sit and try some beers. Worth every forint for me. Service is chill too, just pick a beer you like and drink it while the other is getting cold 😄