Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?
It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Prüfer domain, or equivalently, locally a valuation domain. However, there exist Prüfer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.
If you add the hypothesis that $D$ is noetherian, then this is one of the characterizations of Dedekind rings.