[*]The short answer is: not mandatory in all cases, but it is helpful.
The following article explains when which right of residence applies and which official documents are required for residence (as of 2023).
There are many reasons to apply for a residence card. Depending on the length of your stay, you can get through most of the administrative processes such as opening a bank account and buying real estate in Hungary with a passport from the European Economic Area (EEA) plus possibly a registration card, but there are still some advantages of the residence card.