Both cave and wreck, at least penetrative wreck diving, are both overhead environments, and in so far as this is the case, there will be shared concepts and skills. It is important to note that not all wreck courses are penetrative; or this is merely an option. More on this later...
I took some wreck training. We read the PADI manual and I got a PADI card; it is pretty lacking that manual; but, sometimes having a card is useful when traveling and I did use this card to do penetrative wreck dives in other places.
But, that PADI manual was not the only book we read: all the other manuals we used were cave diving manuals. Laying and following line, lost line drills, blacked-out mask practice, air sharing while moving in confined space with a long hose, etc., all from books written for cave divers. We dove many different wrecks while practicing these skills. We watched videos like this:
(notice, the guy is in an overhead environment, albeit a cave).
There is, then, a Venn diagram to be drawn with an intersecting zone; and, some of the most important and core concepts and skills for both overhead environments are in that intersecting zone. Maybe someone could draw up a chart; this will be more useful than bleating about the different rubber stamps one can get.
I did not just sign up with the first dive shop advertising wreck diving and four wreck dives. I carefully researched who I was going to train with and spent a few weeks with the guy. And, at times, especially at the start, it was good and scary. But, after getting a card, and more importantly, a lot of supervised experience, I was much calmer and more proficient.
I was, a year or so later, diving a wreck while on holiday. There was a wreck dive course going on at that time and the students were on the same boat. I was watching and chatting with the students over the course of a couple days:
The students reported to me that there were no lost line drills; no black mask drills; no long-hose, confined space, blacked-out air sharing or even a silt out 'experience'. No practice diving with or using a redundant gas supply, e.g., a slung pony. They read only the PADI wreck diving manual. They did only 4 dives over two days.
On the last dive of their course there was an optional penetration dive using a finger spool for a line; line practice was just before the dive, on land. Only one of the students was willing to go into the wreck (and this wreck was purposely sunk, upright and relatively new). There was only one finger spool taken inside. So, no backup or tiny spool for a lost line drill. The line, which had a bolt-snap tied on the end, was simply looped through, but not several times around, one of the ship's rusty and crusty stair handles outside the hatch and then clipped back onto itself using the bolt snap. The line was so slack that it both (1) wobbled to and fro in the current and (2) was pulled and yanked and slid around every time the diver moved, creating a risk of cutting the line against the rusty, crusty metal. There was no primary wrap nor a secondary wrap. During the course, the students only dove that same, 'sanitized' wreck; but, to be fair, that was the only one in the area (and this alone would be a sufficient reason, for me at least, to not do a wreck diving course in this place).
That was certainly not how I was trained; I am certain that had I tied a line like that during my training I would have had my ass chewed off - and I was certainly chastised during my training time on a few other occasions - and rightly so too. But, at the end of it, those students had the same 'qualification' card as I, even the students who opted not to go into the wreck and have never run a line before.
So, if there are many 'wreck' dive courses like the one I observed - and I suspect that is more the norm than not from the research I did before deciding where I would train - I could certainly see the argument against the symmetry of skills, in general; but, 'in general' is just that for there are always outliers.
Wreck diving courses are both popular and occur almost everywhere too; even in places with maybe 1 or 2 tiny wrecks. So, there are certainly more wreck divers than cave divers; and this might skew the interpretation of the question of the thread, especially when the content of wreck courses are often not instructed by wreck diving fanatics with a tonne of experience doing overhead or penetrative wreck diving. In other words, 'wreck diver' has a far wider meaning than 'cave diver.' Unless very careful about who one trains one, a wreck course may contain almost none of the core elements required for safe overhead diving, whether that be either cave or wreck diving. Indeed, at the place I am writing about above where I observed the wreck course, the instructor of that course had himself not taken any advanced overhead diver training and had only done penetration dives on that one wreck in that one place since he started teaching the wreck diving course. In other words, he had never entered a wreck with a line prior to teaching that course. I dove with him during my time there and we even did a few penetration dives on that wreck together just for fun. But I myself would not have chosen him to train with him.
I was researching and looking for a fellow to train with for cave diving; but that plan fell through given a change in my job situation. I really don't care about agencies, dive shops with good reviews or certification cards, though the later are useful and necessary to get a ride on a boat sometimes. I primarily look for a mentor with an obvious track record of experience doing what he or she is claiming he can teach me to do and from whom I can gain some experience in that new environment and learn about the quirks of that environment which are not in the intersection zone in the overhead environment Venn diagram. I would think that penetrative wreck diving specialists, and not the mere padi 'wreck diver', are not as unprepared as the bravado of the cave divers might like to think; though, it is more than likely for the standard wreck diver spat out on sort of course I observed to be unprepared. Indeed, some of the cave dives I have viewed online look a lot easier to pass through than some of the rusty, wire-strewn, upside-down and blown-up wrecks I have swam into.
If one is trapped in a silted-out, dark wet hole without a line, sufficient gas, or redundant apparatus, the grim reaper cares not if it is a cave or a wreck.