Similarity between electric displacement and electric field in a linear homogeneous dielectric

Griffiths says

…if the space is entirely filled with a homogeneous $^{10}$ linear dielectric: in this rather special circumstance we have
$$
boldsymbol{nabla} cdot mathbf{D}=rho_{f} quad text { and } quad boldsymbol{nabla}times mathbf{D}=0
$$ so
$D$ can be found from the free charge just as though the dielectric were not there
$$
mathbf{D}=epsilon_{0} mathbf{E}_{text {vac}}
$$.
The explanation given to the same question on this site is that, this is because of the similarity between the below equations $$boldsymbol{nabla}cdotmathbf{D}=rho_{rm f} :::{rm and}::: boldsymbol{nabla}timesmathbf{D}=0$$
$$updownarrow$$
$$boldsymbol{nabla}cdotmathbf{E}_{rm vac}=frac{rho_{rm f}}{epsilon_{0}} :::{rm and}::: boldsymbol{nabla}timesmathbf{E}_{rm vac}=0$$

However how can one deduce from this similarity the relation between $D$ and $E$ knowing that two different functions can have the same divergence and curl,( like two constant functions)?