No, this is not true. For example, all fractals generated from an Iterated function System, (IFS), are self-similar in the sense that the fractal is the union of the images of the fractal itself, under several maps. See Hutchinson operator.

The classical Julia fractals can be interpreted in this manner, where the two branches of the inverse of $z \to z^2+c$ are used.

Then, there are the random midpoint displacement fractals which are self-similar in a much weaker sense than above, since these fractals involve random choices.

Not all fractals are given by iterating a map (I prefer to call these types of fractals discrete); some fractals are constructed using continuous methods, such as the Lorenz attractor. But note, this fractal can be approximated using iteration of a set of maps.

No, this is not true. For example, all fractals generated from an Iterated function System, (IFS), are self-similar in the sense that the fractal is the union of the images of the fractal itself, under several maps. See Hutchinson operator.

The classical Julia fractals can be interpreted in this manner, where the two branches of the inverse of $z \to z^2+c$ are used.

Then, there are the random midpoint displacement fractals which are self-similar in a much weaker sense than above, since these fractals involve random choices.

Not all fractals are given by iterating a map (I prefer to call these types of fractals discrete); some fractals are constructed using continuous methods, such as the Lorenz attractor. But note, this fractal can be approximated using iteration of a set of maps.

New Edit: The Mandelbrot fractal cannot (most likely) be constructed via IFS, (I have asked this question before here on MO).

No, this is not true. For example, all fractals generated from an Iterated function System, (IFS), are self-similar in the sense that the fractal is the union of the images of the fractal itself, under several maps. See Hutchinson operator.

The classical Julia fractals can be interpreted in this manner, where the two branches of the inverse of $z \to z^2+c$ are used.

Then, there are the random midpoint displacement fractals which are self-similar in a much weaker sense than above, since these fractals involve random choices.

No, this is not true. For example, all fractals generated from an Iterated function System, (IFS), are self-similar in the sense that the fractal is the union of the images of the fractal itself, under several maps. See Hutchinson operator.

The classical Julia fractals can be interpreted in this manner, where the two branches of the inverse of $z \to z^2+c$ are used.

Then, there are the random midpoint displacement fractals which are self-similar in a much weaker sense than above, since these fractals involve random choices.

Not all fractals are given by iterating a map (I prefer to call these types of fractals discrete); some fractals are constructed using continuous methods, such as the Lorenz attractor. But note, this fractal can be approximated using iteration of a set of maps.