There's no One True Definition of what it means to be fractal, but a lot of working mathematicians use the criterion that a space's Hausdorf dimension is not equal to its topological dimension. Oftentimes (but not always), the Hausdorf dimension will be fractional, which is where the word comes from. Self-similarity is an easy way to satisfy that requirement in a way that's easy to explain, but it's far from the only way.
I was under the impression that the fractal dimension must be greater than the topological dimension for the space to be a fractal.
And that the fractal dimension is usually equivalent to the mincowski dimension (the limit of the area measured by finite boxes as the size of the boxes grows arbitrarily small) and hausdorf dimension (the limit of the perimeter as measured by an aproximting polygon with equal length sides as the number of sides grows arbitrarily large) along with others.
