I just posted a question about English etymology in Linguistics SE.
I was contemplating whether to post it here, but decided against it.
Could I have posted it here, or would it have been off-topic?
I think NVZ is right: it may very well have been closed on EL&U, sadly. This site is very close happy about questions that are very interesting but that is not exclusively about modern English (and also about other types of questions). I would certainly support your question, though.
The cause of this that it takes the votes of only 5 high-rep users to close a question (a tiny minority of high-reps), and the other high-reps can do nothing whatsoever to prevent this from happening if they're just browsing the site, however much they like the question.
The only thing they could do is try and find the question in the Review Queue and vote to leave it open, but that is a rather tedious exercise, and it can only be done after the question has already received close-votes. So there is no way to vote "don't close" during normal question-browsing until after a question has already been closed—at which point the question has been up for a while, and most users have already seen it once, and they are unlikely to view the question again later. The number of high-rep users is ever increasing, so this problem is unlikely to ever be solved.
Since both words are (now) part of the English language, I think that asking about their etymology would be on-topic at this site. Also, the question you posted at Linguistics SE shows research effort on your part, so the question would clearly have passed muster on that account.
Just by way of underscoring that no easily accessible answer to your question is available online, you might have added to the question (if you had asked it at English Language & Usage) that you had consulted Etymology Online (the etymology resource of first resort at this site) and found no hint there that angst and anxiety are related.
At that point, the only threat to your question would be from question monitors who might feel that going farther back than Old High German and Latin in search of a shared ancestor is too distant from the Englishness of the two words to justify posting the question on EL&U. I disagree: To me, welcoming questions about etymology (which we do) means welcoming questions that look back deep into the past—and I think voting to close your question because it might require delving as far back as Indo-European would be unreasonable. But some of my fellow EL&Uers take a very narrow view of "English language and usage," and I can't (and don't) speak for them.