Pairwise hash functions that are independent from each other
If you select $h_1,h_2$ independently at random, then by definition

$$\Pr_{h_1,h_2}[h_1(x)=y \land h_2(z)=w] = \Pr_{h_1}[h_1(x)=y] \times \Pr_{h_2}[h_2(z)=w].$$

If your hash family is universal, then both probabilities on the right-hand side will be $1/|D|$.

In other words, if $H$ is universal, then it automatically follows that

$$\Pr_{h_1,h_2}[h_1(x)=y \land h_2(z)=w] = 1/|D|^2.$$

Every universal collection works -- no extra conditions or requirements are needed.