I'm afraid I havn't visited the place myself, so I can't give any subjective pointers regarding the museum. I also don't know much about the Arrow Party, so I'll leave that to someone else - you may even poke one of our users with relevant flairs (/u/depanneur perhaps) for that information.

The House of Terror is a site museum, meaning that it is located - in a reconstructed state - on the same spot as the original house. It is (or was), as the name suggests, a place with a sole purpose of inflict fear in the citizens of the city. If the black car came to pick someone up and took them to the house, they were never seen again.

This is of course rather different from the death camps, whose purpose was to kill people, to get rid of them. People were brought to Terror Haza to be interrogated, tortured and then killed. Simple enough. The purpose of the museum isn't mainly to show that though. It is hardly anything special about such an institution, especially with all the Holocaust museums that have sprung up around the world.

The museum exists to show upon the people who were responsible for what happened there. If you've read Christopher Browning's Ordinary Men, you may recognize the way of thinking. In this museum there are walls full with pictures of people, presumably the victims that died there. However, some of these people are completely anonymous, while some others are actually given names, as well as birth and death dates. These are not victims, these are the people who worked here.

That's where the powerful message comes through. The people on these pictures aren't the likes of Adolf Hitler or Josef Stalin, they are ordinary people. People who would wake up in the morning, go to work, change to their work clothes and commit atrocities; but when the work day was over, they would put on their regular clothes and head home. They were your neighbours, the parents of your children's friends, and so on. That brings the message from being about "they did this, we should have stopped them" to "we did this, how could we?".

The obvious answer to that is that someone else ordered it. The Nazis argued that way during the Nuremberg trials, someone else ordered the killings, the Holocaust. They were just the tools that did the deeds, they held no responsibility. The point of the House of Terror is to make the onlooker question the heritage itself, to change the keyword from "they" to "we".

Which brings us to your question: "How does Terror Haza affect the identity of Hungarian people today?". It does not, as one may hope, actually reach through in the way it should. The majority of visitors to Terror Haza aren't in fact Hungarians, but tourists; it is an issue with two probable explanations.

First, and relevant to the rest of the post, you have the feelings this museum create. Feelings of guilt or uncertainty, that make people unsure about their place in history and about how they ought to view their deceased relatives who were part of these deeds. However, that is just one reason, because, secondly, we know that people who visit museums are in a large majority people who are well educated, middle class and up. Sadly, the people living in Budapest, because of Hungary's societal problems, more often than not have better things to do than to visit a museum.

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Further reading:
Aronsson, I-L. (2009), Heritage: A Conceptual Paper, Toward a Theory of Cultural Heritage in Humanitarian Action.

Duffy, Terence M., (2001). “Museums of `human suffering' and the struggle for human rights”, in Museum International (UNESCO, Paris), No. 209 (Vol. 53, No. 1, 2001).