I live in Croatia now but before the lockdowns etc. I used to often go to Hungary for shopping trips or to meet friends. Early in September I attempted such a shopping trip but it did not go well as I managed to lose my temporary residence card. I had only included the card with my passport in the hope that my British passport would not get stamped if they were aware I live in an EU country. My passport did get stamped and I did complain. I was told they have to stamp it because I'm coming from a non Schengen country. As UK was never in the Schengen zone I can't see what difference that would make.
I had to apply for a new card which took a few weeks. In the meantime my Hungarian bank debit card expired and a new one arrived by post. When I tried the card in an ATM in Croatia it didn't work. I thought perhaps the first transaction to activate the card must be made in Hungary so today I visited Hungary to test this idea. It didn't work so I will have to write to the bank and ask what to do about it.
This time I only gave my passport and vaccination certificate at the Croatia side of the border but included the residence card at the Hungary side. Again they stamped the passport. On the way back to Croatia a couple of hours later I didn't give the residence card at the Hungary side but they asked to see it! I don't know why they needed it unless it was curiosity to check I had got one.