Q:

Was Sergei Brin a refugee?

According to the World Economic Forum:

5 refugees who changed the world
...
Sergey Brin
He might be one of America’s most famous entrepreneurs, but Sergey Brin was not actually born in the US. In 1979, with the help of the Hebrew Immigrant Aid Society, Sergey and his family managed to leave the Soviet Union, where they’d been facing growing anti-Semitism.

In addition the man himself recently joined protests at San Francisco airport, where he said that:

But Forbes’ Ryan Mac did catch Brin elaborating slightly — reportedly saying “I’m here because I’m a refugee.”

Was Sergei Brin a "refugee"?

A:

There are two aspects to this question:

Did Sergei Brin's family enter the US on a refugee visa?
According to an article in the New York Times, Sergei Brin considers the Hebrew Immigrant Aid Society to be instrumental in his family's arrival to the US. Their website contains an entry about his parents, which links to a lengthy interview with the Brin family about their history. The HIAS entry mentions that:

Our wait for the permission took nine months of stress and uncertainty. We (I, my husband Michael, our six year old son Sergey, and my mother-in-law Maya) left Moscow on June 15, 1979. We discussed a possibility of going to Israel, but after all decided to start our new life in the USA.
While we waited for our American visas in Europe, Michael received an offer of a visiting professor position from the University of Maryland. That is where we finally arrived - in College Park, Maryland.

And according to an article from the Center of Immigration Studies:

It referred Jewish emigres not wanting to go to Israel ("dropouts") to the American Joint Distribution Committee (JDC) and the Hebrew Immigrant Aid Society (HIAS) for assistance to go to other countries. These organizations moved the emigres to Rome where HIAS assisted them in getting visas and JDC provided shelter and a subsistence allowance.

The article goes on to say this about visas:

By the summer of 1979, the situation in Rome became more problematic. Only 11,000 of 25,000 allocated parole visas for the United States had been used; the remaining 14,000 had to be used by the end of September 1979. HIAS, however, processed only 700 persons per week.

The answer is therefore inconclusive: it seems that most Jews entered the US under a "parole visa", which may not be a full equivalent of the refugee visa.

Was Sergei Brin a refugee, as currently defined by the UN?
According to the UN Refugee Agency:

A refugee is someone who has been forced to flee his or her country because of persecution, war, or violence. A refugee has a well-founded fear of persecution for reasons of race, religion, nationality, political opinion or membership in a particular social group. Most likely, they cannot return home or are afraid to do so. War and ethnic, tribal and religious violence are leading causes of refugees fleeing their countries.

While Brin's family was in fact discriminated against during the Soviet times due to their Jewish status, their livelihood seems to have to been average by Soviet standards:

The Brins’ encounters with institutional anti-Semitism did not extend to day-to-day interactions with colleagues and neighbors. Highly assimilated into Russian culture, they were part of the intelligentsia and had a circle of university-educated friends. Occupying a tiny, three-room apartment in central Moscow, 350 square feet in all shared with Michael’s mother, they were better off than many Muscovites who still lived in communal apartments. 

And Brin's parents weren't forced to quit their job until they applied for a Soviet exit visa:

They formally applied for an exit visa in September 1978. Michael was promptly fired.

Therefore the answer to the second question seems inconclusive to me. While the US did accept all Soviet Jews as refugees, not all of them fall under UN's definition of who a refugee is. It should also be noted that once in Vienna, all Jews could travel to Israel instead and automatically receive Israeli citizenship, rather than becoming a refugee elsewhere. The Center for Immigration Studies seems to agree:

Most Soviet Jewish Zionists who only wanted to immigrate to Israel had done so by 1973. The overwhelming majority leaving after 1973 were motivated more by economic betterment than by Zionist ideology. They saw Israel as a very small market with fewer opportunities.

