How do you navigate your expat identity in Hungary?

Yes, you will always be the "outsider" it is true.
My husband lived in the USA for more then 40 some years and daily was asked where he came from.
People are just that way with anyone slightly,"different" then them.
Your personality is totally different in a new country as well, most people do not understand the different words plays or jokes one tries to share.
After a bit you just learn it isn't worth the hassle to talk to everyone or to bother with just everyone either.
Not much different then in ones home country really, not buddies with everyone one runs into.
As it takes many years to learn the in's and out's of a new language you often will come off as hard, dull or as a new person with your new language.
I know in English in the uS my HU husband was ot=ften perceived as hard, super cold and having not much humor.
All untrue as he is kind, funny and carrying in reality.
Yes, you will be lonely at first if you need people to validate you.