There is difficulty in defining in what a fractal is because there are many fractal dimensions (boxcounting, correlation etc.).

Quite interesting.

Is something still a fractal when there exists a fractal dimension which is equal to topological dimension and a fractal dimension which is not equal to topological dimension?

🧐 Why does this sound contradictory. Or are you talking about an object with 2 topological dimensions and above?
Probably in the first dimension of, say a 2D object, there's self-similarity but it's topological dimension is not the same as the fractal dimension and in the second dimension there's also self-similarity but the topological dimension is the same as the fractal dimension. Or is there something I'm missing ?

If it's for the 2D object, then i don't know. Plus i would love to see a visual representation of this kind of pseudo-fractal - if that's the right word to use.