How to find out someone really holds a Doctorate in Electrical Engineering from Hungary? How to find out if someone holds a Doctorate in Electrical Engineering from Technical University of Budapest? 
I know that the university changed its name to Budapest University of Technology and Economics.

He claims he got a Doctorate in Electrical Engineering from that university in 1972. What I know is that he has a professional "engineering degree" from that university registered in the Engineer Association in my country. But the translation of his degree does not say Doctor nor PhD, just electrical engineer. The certificate is in Hungarian with a translation to Spanish, but translation has no word of "Doctor" anywhere.

Another thing is that I find no record of any article nor paper nor publication from him on any journal. Well obviously I can not look for in Hungarian language.

When I finished my PhD (in math), I was registered in math genealogy project and I have one paper published on a reputable math journal.

Is there a web-page similar to Math Genealogy Project for electrical engineering?

I have also written to the university but have no reply until now. I do not know also if Hungarian Universities may release records of former students.

Is there anyway to verify that person really holds a doctorate? Or how can I prove he does not have it?
You could try writing to someone who teaches mathematics there --- see the course coordinators listed here. Pick someone who is likely to have heard of you (or your Ph.D. advisor, or any of your co-authors, or someone who might be marginally interested in anything you've worked on) and make sure the email subject line reflects this connection. I've done this sort of thing MANY times, even a few times in the early 1990s by postal mail. Also, try writing to a reference librarian, something I've also done successfully several times.

To hopefully not overstate the obvious, if you write to someone in mathematics (a field I chose because you said you are in mathematics), this person is almost certainly not going to know the answer to your question, but he/she should be able to assist you in locating someone who can investigate your question, and you should be clear about this in your email to any such person in mathematics.