"Press flesh" is commonly used to refer to a situation where individuals shake hands, socialize, or establish contact in a political or social setting. As a synonym, one may use the phrase "to rub shoulders," which means to interact or associate with important or influential people. Another synonym could be "networking," which is the act of establishing or maintaining professional relationships. Alternatively, "meet and greet" is another synonym that can be used in a similar context to press flesh. In essence, the phrase refers to any situation where individuals come together and establish a connection either for business or social reasons.