I don't agree that one's culture defines a person, but it certainly exerts significant influence upon a person's development. One simple example that appeals to me is to look at musical preferences. Those of us who grew up with Western European-based musical training are accustomed to hearing major and minor scales and chords derived from those scales, all of which have specific recognizable tonal intervals that sound "right" because of the experience and training we have had with the music surrounding us. The tonalities of music from the Far East strike us as very dissonant - because we have not been exposed to those combinations of sounds enough to learn to hear, accept, and appreciate them.

One's culture does define one, although this is a debate that rages in many academic fields including Linguistics. The preponderance of opinion now leans toward the concept that language defines one's perception of reality and therefore one's perception of oneself. Therefore, since the particulars of language are cultural, culture defines one. An example of this is Yiddish speaking Hebrews in Russia. Their daily language was not pure Hebrew nor was it  Russian. Their culture founded a new variety of language that accommodated the particulars of their culture and defined how they perceived reality, thus defining them. This is a good example of a common experience because Yiddish is not a standard language thus a simpler example to understand.

The writer W. Somerset Maugham felt that culture creates the person. He wrote,

It is very difficult to know people.  For men and women are not only themselves, they are also the region in which they are born, the city apartment or the farm in which they learned to walk, the games they played as children, the old wives' tales they overheard, the food they ate, the schools they attended, the sports they followed, the poets they read, and the God they believed in.  You can know them only if you are them.

Indeed, one's environment, one's nationality, one's race, one's religion--all these form much of one's essence since some of the traits are in the DNA.  This is one reason that immigrants who came through Ellis Island moved into the neighborhoods that housed those of their own culture. These people were defined by their religion, the foods that they ate, the types of clothes that they wore, the colors they liked, the values that they had, the customs that they held, the type of hairdos and mustaches or beards that they wore, etc.  In short, Greeks were easily identifiable, Italians and Irish, Lithuanians, etc. were all recognizable by their mustaches, clothes, mannerisms. foods, etc.

Now, in America and in some other countries, many nationalities and races are so mixed that these defining notions are not always apparent.  And, few cultural patterns and customs are followed since there is no single culture to be defined.

Values and attitudes are often generated by one's culture, as pointed out above. These are related to perspective. We see the world through a particular lens; filter our experience through a particular set of ideas. This lens and these ideas are functions of our cultural background.

However, "culture" can be understood as both a local and as an a-local phenomenon. We can talk about a culture of the home and a culture of the community. Both will shape a person's perspective and  in that way define a person's identity. 

When you grow up in a certain culture, you tend to reflect the values of that culture. That doesn't necessarily mean that everyone in that culture thinks exactly alike, of course, but there are certain broad parameters that most of that culture's people would adhere to.

In the United States for example, we stress the value of individual freedom above most other things. This is probably not the case in countries that exercise greater control over their citizenry.

Keep in mind that there may be many cultures in any one country, so there are important differences between those groups.

It has so much to do with who we are because it affects a number of aspects of our attitudes.  For example, I grew up in a place where making eye contact with superiors is rude.  This strongly affects how I behave today.  However, not all aspects of culture do stick with us.  I wear shorts almost all the time even though wearing shorts was seen as somewhat improper in the culture where I grew up.

I agree that culture does not define a person. Instead, culture can influence who a person becomes. In that, some people fail to understand their own cultures and leave (think about all of the people who come to America to escape conflict). Therefore, based upon this example, some people staunchly dismiss the place of their birth and that culture given they do not agree with the ideologies presented.

A person defines him/herself alone. While many different things can influence them (parents, community, ideologies, morality, religion, culture), a person becomes who they are based upon many different attributing factors.

I have always said my culture is part of who I am, but it doesn't define who I am.  I think a person's culture is definitely an influencing factor in their life, but their own thoughts and actions define their individuality.