The symbol is the representation of the reality that we perceive by the senses, and associate an idea with the colectives imaginary.
Y don´t really think that all the designs have symbols, but I personally like to use symbols to create more complex and/or expressive messages. Symbols are good tools for this.
But not only the objects are symbols, also the abstract elements are. So on, a curve normally represents the nature, classic things or humanities, instead of a square, that is more a representation of machines, industrial age or science. It is not the figure the only thing that make us realize that the man is a man in Klims paint “the kiss”, but the rectangles in his clothes, as well as the circles of her dress, are women symbols, represented throught the curve.
Similary, we can see in the “crying spider” of the symbolism genius Odilon Redon that he uses the spider symbol to represent himselve. But it is not the same a colorful spider than this spider. I mean that abstract symbol are complex and very expresive. In this paint the darkness express even more than the spider.
In design, every curve is a desition that we take, and it symbolizes something related with the real world. Perhaps what makes a symbol what it is, is the intention to be a symbol.