We study diffusion in heterogeneous multifractal continuous media that are characterized by the second-order dimension of the multifractal spectrum , while the fractal dimension of order 0, , is equal to the embedding Euclidean dimension 2. We find that the mean anomalous and fracton dimensions, and , are equal to those of homogeneous media showing that, on average, the key parameter is the fractal dimension of order 0 , equal to the Euclidean dimension and not to the correlation dimension . Beyond their average, the anomalous diffusion and fracton exponents, and , are highly variable and consistently range in the interval [1,4]. can be consistently either larger or lower than 2, indicating possible subdiffusive and superdiffusive regimes. On a realization basis, we show that the exponent variability is related to the local conductivity at the medium inlet through the conductivity scaling.