I checked the 5/50 type 1914....the standard DD gun for IJN destroyers. According to Campbell "Nose fuzed HE of two different types was carried and there were also two types incendiary shrapnel and two types illuminating shrapnel and the ASW round again.
So greater choice ion the 5in calibre, but neither was intended as an AP or SAP round.
Naval shells use different specifications and designations instead of AP/SAP/HE.
They indeed have AP and HE designations, but not SAP. And probably the closest thing to SAP is what they call a "common" round, which is in essence a HE round but with thicker walls designed to penetrate lighter armor.
As a real rough rule of thumb, "Common" rounds are designed to penetrate 1/3 their diameter, so in the case of a 5" shell it should penetrate 1.67". There are so many variables here, so it indeed is a real rough estimate. Muzzle velocity of the gun, range, butt or base fuse, angle, hitting near a joint are just a few of the many variables. But this penetration should penetrate most destroyer armor. It is said that destroyers have no armor, but this is incorrect, it's usually around 1" or 25mm or so of thickness, so a 5"Common round will generally have enough to penetrate a destroyers armor.
These different designation's are compounded by the US use of a "Special Common" round, which is a little but thicker walls and a smaller bursting charge. This round gives the US 5"/38 gun similar performance to the Japanese 5"/50 when it comes to penetrating armor - at the expense of a smaller bursting charge.
It gets a bit confusing when it comes down to caps and wind screens, I know the US common rounds did not have either, not 100% sure of other navies. And I thought, but I could be wrong that common rounds were butt fused. Perhaps it is the Special common round that is butt fused.
But the point of all this is that AA rounds were high explosive, and the preferred round for surface to surface combat was the Common round or Special Common. AA or HE rounds had a larger bursting charge than common rounds, though thinner shell walls and correspondingly less penetration.
And my point behind all this is that the 3.9" had a HE/AA round, but not a common round designed for it, while the 5"/50 did have common rounds.
Some publications do not differentiate between Common and standard HE. So you don't think I made these up
, here is a good guide for shell designations:
Definitions and Information about Naval Guns - Part 2
and a special ASw round (i confess i dont know what that is).
A blunt "diving shell" type of round, used obviously against subs, but I'm not sure of it's sucess. The Japanese had researched "diving shells", as far as I know other no other navy did anywhere this degree of research. They had some designed for surface to surface as well, and this is one reason why later Japanese battleships had a much more substantial lower belt than other countries battleships.
Shells generally have a tendency to "skip" if they have a relatively flat trajectory, the diving shells were modified not to skip but to dive. My guess is the ASW shells are similar.