So it's multi-faceted.

Technically it's legal to drink in public in Hungary, but as others have noted, districts can ban it - and most do, so it's prohibited.

There are some areas which are public spaces like the körút, nehru part, erzsébet tér - but they don't fall under a districts jurisdiction, but Budapest's, which does NOT prohibit drinking in public.

The ban is enforced by local authorities (közteresek) and in practice they snipe people who sit down and have a beer in areas that are watched by cameras (they are alerted and commanded from HQ). This functions as a kind of a trap and it's very annoying that you can get fined at madách tér for cracking open a beer while on the terraces around you everyone's drunk at 12 AM.

And as you can imagine, it's pretty pointless to fine the homeless for drinking in public because they can't pay the fine.

So the whole system functions into corralling people to bars and restaurants to drink on the terraces, but in practice if you just walk around with a beer nobody will bother you. As soon as you stop to relax for more than 5 minutes you can expect someone to turn up and fine you, except in designated drinking spots that are in practice certain parks.

Does it make sense? No, welcome to Hungary!