Is there a formal and informal way of speaking in Hungarian?
Yes there is. I couldn't find anything better: https://en.wikipedia.org/wiki/T%E2%80%93V_distinction#Hungarian. Running through the wiki article on Hungarian (https://en.wikipedia.org/wiki/Hungarian_language) might put a few things in perspective.
Yes, there is, exactly the same as in German (du / Sie) or Italian (tu / Lei). The only difference is that in Hungarian there are two ways to say "Sie": "ön" (more polite), and "maga" (less polite, and nowadays getting somewhat rare). Grammatically, "ön" and "maga" are totally the same. "Du" is "te". The informal way is called tegezés (derived from "te"), the formal way is önözés/magázás (derived from "ön"/"maga"). In everyday life, even in working environment, nowadays it is very common to use the informal way exclusively. The formal way is used eg. in government offices, at the police, at the doctors, in most of the shops, restaurants (but in fast food restaurants they speak informal). You will get a feel when to use which after spending a few months in Hungary.
It's mostly just the change of the conjugation (Te -> Ön, Maga) but also there are words that are usually only used in formal speech that replace informal words (krumpli -> burgonya) and a little grammar, like avoiding the passive voice.
There are also more polite ways to talk. For example when you want something it is nicer to say “szeretném” instead of “akarom” which is a more demanding word. There are synonyms in every language that have nuances of meaning.