This is a reasonable candidate for a notion of a "Fractal" but these days it is generally accepted that there is no universal definition.

For instance, this definition would not include objects such as the Devil's staircase [1] or more generally images of the unit interval under a continuous monotonically increasing function.

Some more exposition about attempts to rigorously define the notion of a fractal can be find in the introduction to Kenneth Falconer's book [2]

[1] https://en.wikipedia.org/wiki/Cantor_function

[2] https://zbmath.org/1285.28011

Edit: Fixed incorrect zbmath link.