Are you referring to national or indigenous cultures?
National:
both in Mexico and in South America the leading language is Spanish, and the countries have a history of being a colony of Spain, which influences the culture. Also, the majority of the inhabitants are Mestizos, and the culture is influenced by the indigenous cultures of America, which further causes many similarities.
Indigenous:
Already indigenous cultures of Mexico and Southern American were similar as their gods are similar (such as the God of Sun), they had ritual sacrifice and through the trade and common natural resources, their cultures shared many other similarities.