The expression is properly “in the same vein.” The definition of vein as given here is:
I believe the usage in the expression would come from parts 2 and 3 above, where the vein is a pathway, the idea being that the thought being introduced by the expression “in the same vein” takes the same or a similar pathway as the referenced idea. I can’t imagine how “vain” would be worked into the usage.
And I’ll note that a google search on both phrases turns up quite a few “vain” usages, none of which make much sense. I suppose, maybe, if you were comparing a couple of prima donnas…
A useful resource for this sort of question is the Collins Cobuild Concordance, a searchable database of English sentences used as the basis of their dictionaries: search it for “in the same vein|vain” to match both forms, and it has nearly 20 uses of “vein” and none of “vain”.
(1) A vein is a blood vessel or a distinctive style.
(2) Vain means ‘self admiring’ or ‘futile’. The term ‘in vain’ means ‘without success’.
(3) A vane is a stabilizing fin or blade.
Since this zombie has risen from the grave, it’s worth pointing out that Google gives 13M results for vein vs 270K results for vain. Either the OP didn’t do a very good job of searching, or we are witnessing the march of progress before our very eyes.
I’m 63 and only heard or used the term “buck naked” since childhood. The internet is full of speculation as to where the term came from, including that it may have originally morphed from “butt naked”, which doesn’t seem likely. I never heard “butt naked” until recent years.
Ah crap, I’m talking to a zombie.
The original form (and still the correct one, at least here in the UK) is “buck naked”. It seems to have mutated in recent years, in the same way that “off your own bat” is often mangled into “off your own back”.