A Poisson distribution is one with a PMF of the form $\exp(-\lambda)\frac{\lambda^k}{k!}$ for some $\lambda>0$, its support the set $\Bbb Z^\ast$ of non-negative integers. In particular, $\sum_{k=0}^\infty\exp(-\lambda)\frac{\lambda^k}{k!}=1$ regardless of $\lambda$. The probabilities sum to $1$, of course; that's what we call unitarity. I'll use the term "accidental unitarity" for the superficially similar-looking result $\int_0^\infty\exp(-\lambda)\frac{\lambda^k}{k!}d\lambda=1$, regardless of $k$ (in fact, it works for any real $k\ge0$).
There are distributions that are "nearly accidentally unitary", if you pick the right integration range. For example, the Poisson distribution is obtained via $n\to\infty$ from $B\left(n,\,\frac{\lambda}{n}\right)$, so it makes sense to look at something that happens with Binomial distributions:
$$\int_{0}^{n}\frac{\Gamma\left(n+1\right)}{\Gamma\left(k+1\right)\Gamma\left(n-k+1\right)}\left(\frac{\lambda}{n}\right)^{k}\left(1-\frac{\lambda}{n}\right)^{n-k}d\lambda=\frac{n}{n+1}$$(a trivial exercise in Beta functions), again regardless of $k$.
But I doubt we can write a similar result in general. For example, the logarithmic distribution relies on the identity $\sum_{k=1}^\infty\frac{-1}{\ln(1-p)}\frac{p^k}{k}=1$ for any $p\in(0,\,1)$, but $\int_0^1\frac{-1}{\ln(1-p)}\frac{p^k}{k}dp$ depends on $k$ in quite a complicated way (which I've still not fully ascertained; but based on small $k$, for any $k\in\Bbb N$ it seems to be a rational combination of the logarithms of finitely many positive integers).
Is what happens with the Normal and Poisson distributions as meaningless in isolation as the sophomore's dream, or is there more to it? For example, can we identify a sufficient but not particularly stringent condition for a one-parameter family of PMFs $f(\lambda,\,k)$ viz. $\sum_{k\in S}f(\lambda,\,k)=1$ with the same support $S$ to yield a $k$-independent integral $\int_{S^\prime}f(\lambda,\,k)\lambda$, ideally equal to $1$? (Or to mirror the binomial case, $\sum_{k\in S_n}f_n(\lambda,\,k)=1,\,\lim_{n\to\infty}\int_{S_n^\prime}f_n(\lambda,\,k)\lambda=1$.)
I suspect my decisions on how to tag this question could do with some work, so feel free to edit those.