We study diffusion in heterogeneous multifractal continuous media that are characterized by the second-order dimension of the multifractal spectrum D2, while the fractal dimension of order 0, D0, is equal to the embedding Euclidean dimension 2. We find that the mean anomalous and fracton dimensions, d(w) and d(s), are equal to those of homogeneous media showing that, on average, the key parameter is the fractal dimension of order 0 D0, equal to the Euclidean dimension and not to the correlation dimension D2. Beyond their average, the anomalous diffusion and fracton exponents, d(w) and d(s), are highly variable and consistently range in the interval [1,4]. d(w) 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.