Hey Beatrycze!
That was a very amazing question... however not easy. I made a small background check in the topic to understand the artist and the statue too.
The customer of this statue was the Ministry of Education and Religion of Hungary, they ordered this statue for a woman, called Irma Krail (later Báró Strallenforf Brúnóné).
Irma had four husbands, was extremely rich and had no children at all. So after her dead she gave her houses as a heritage to the Ministry for charitative purposes, especially for education. So for showing some respect the Ministry payed her a cenotaph. This one.
The artist was Ligeti Miklós.
Ligeti had fauns and other antique characters in his portfolio. He had some allegoric and "zsáner" elements to, but mainly he used these antique and hellenistic cliches and never the elements of the Hungarian mithology - which was not part of the artistic tradition in this era.
Of course, these was the years of the millenia, so the search of our origins, the past and the Hungarian identity was a popular and vivid theme for the Hungarian artists too, but mostly in the painting. Not in the statues.
So, consequence:
This statue represents a woman - whose face model was the dead woman, Irma herself. My resources suggested that the animal is an owl. Just an owl. And I think it could be true. The owl symbol is just for the sake of the Education Ministry. Because of the symbol's original meaning and the will of the dead women (charity for education).
So no demonettes, catlike birdish creatures here. This statue has a bronze version to... maybe the material forced and shaped this owl to look like a bit... odd.
Our old shamanic and later christian "mythology" has no catlike but owl shaped creatues. As I know so far. We had so many creatues, Turul (eagle), Fehérlófia (horse-like man hero character), Csodaszarvas (stag), wolf-like creatures, so many tree and floral motifs, bad insectoid things, snakes living at the roots of the Life Tree... but I do not know any cat-owl figures in our original traditions.
However - I hope you loved your trip to Budapest!
And I am happy that you remembered me and asked me this question. Yeah, that was interesting. ^^ Thanks! : )