What you are calling $\pi^2(N)$ should really be $\pi^1(N).$ It is slightly unfortunate that this is the number of twin primes up to $N$ which is commonly written as $\pi_2(N),$ but that is just notation. There is no reason to think that $D$ being prime will make a difference. It will be clearer to let $D$ be *any* nonegative integer and

$$\pi^D(N)=\begin{cases}
\vert [p: p\leq N, p\in\mathbb{P}, Dp-2\in\mathbb{P}]\vert& \text{if $D$ is odd},\\\
\vert [p: p\leq N, p\in\mathbb{P}, Dp-1\in\mathbb{P}]\vert& \text{if $D$ is even}.
\end{cases}$$

For "large" $N$ relative to $D$ there will be hardly any difference between the density of primes near $N$ and near $DN:$ $$\frac1{\ln(N)}-\frac1{\ln(DN)}<\frac{\ln(D)}{(\ln{N})^2}.$$
What should really matter is the set of odd primes dividing $D$ (the prime $2$ is built in.)

Recall that the number of primes up to $N$ is $\pi(N)\sim \frac{N}{\ln{N}}.$ It is quite true that no one (yet) can prove that $\lim_{N \to \infty}\pi^1(N)=\infty$, but it is known how fast it goes there. Less humorously: it is widely believed that the number of twin prime pairs up to $N$ is $\pi^1(N)=\pi_2(N)=C_2\frac{N}{(\ln{N})^2}$ for the constant $C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016.$

We can discuss the present question with a high confidence that computational results will support us. In the spirit of the amazing paper Heuristic Reasoning in the Theory of Numbers (read it!) one might define $R_D(N)=\frac{\pi^D(N)}{\pi^1(N)}$ and experiment to make conjectures. I will predict that for large enough $N$ $R_2,R_4,R_8 \sim 1$ , $R_3,R_6,R_9,R_{12},R_{18} \sim 2$
and in general $R_D$ will be very close to the $R_d$ from Polya's paper. This will take very large $N$ for larger $D.$ The same phenomenon should be easier to observe for $\pi^D(M,N)$ with the condition on $p$ changed to $M \le p \le N.$

Here is a very small experiment. The first $2000$ primes following $10,000,000$ run from $M=10000019$ to $N=10032181.$ The formula number of twin primes predicts $\pi_2(N)-\pi_2(M)=143.27.$
Here is a sorted list of $[D,\pi^D(M,N)]$ for selected values.

[3, 329], [6, 305], [9, 296], [24, 282], [12, 280], [18, 276], [27, 259], [36, 242], [5, 218], [10, 188], [25, 182], [20, 176], [1, 169], [2, 166], [40, 163], [16, 152], [4, 140], [8, 137], [32, 135]

The values are grouped as predicted. The values for $1,2,4,8,16,32$ were supposed to all be around $143.27.$ The evidence given is fair but maybe not overwhelming. Here are the ratios $\frac{\pi^D}{143.27}$

[3, 2.295882763], [6, 2.128401954], [9, 2.065596650], [24, 1.967899512], [12, 1.953942777], [18, 1.926029309], [27, 1.807397069], [36, 1.688764829], [5, 1.521284020], [10, 1.311933008], [25, 1.270062805], [20, 1.228192603], [1, 1.179344033], [2, 1.158408932], [40, 1.137473831], [16, 1.060711793], [4, .9769713887], [8, .9560362875], [32, .9420795534]

I leave more extensive calculations to the motivated reader.