I have revised your question a bit because I believe you are inquiring about whether metaphors are culturally specific, not "culture or specific." The answer to your question is "yes" and "no." There are many universal metaphors, and there are metaphors that are culturally specific.
First, let's talk about some universal metaphors. Metaphors that rely on the idea that "up" is good and "down" is bad are pretty much universal. For example, in English, we speak of the stock market being up or down. In other languages, one's financial fortunes are reflected in similar ways. Even if one's financial worth is reflected in clam shells or gold, a large pile is higher and more "up." This seems to be a universal attribute to wealth. In English, we also tend to speak of mood as being "up" or "down." And this is universal, as well. Think about your body. If you are not happy, you tend to be slumped over, literally "down." If you are happy, you tend to be standing tall, and your body literally feels more "up." This is a human tendency,so it doesn't matter what culture you are from.
Other metaphors are culture specific, and idioms are a good example. In English we say it's raining cats and dogs. In a culture in which cats and dogs are not pets, such a metaphor would not be understood. Other examples are metaphors about business. In a bartering culture, the idea of exchange must be expressed differently than in other kinds of cultures. In English, we speak of a place that manufactures things as a plant, but you can see that that metaphor would not work in a culture that had no manufacturing.
There are people who are anthropological linguists who study this universality and the differences among cultures in metaphor. One such expert is Zotan Kovecses, who has written numerous books on similarities and differences among cultures. Another person who writes frequently about metaphor is George Lakeoff. I have provided links to more information about both authors.