This is the most hilarious bug I've run into in a while. It sure is a top contender for one of he best, but at least I managed to reproduce it now thank to @Azdeus's testing.
Steps to reproduce:
Be from Commie-EU-Yurop so the cookie settings pop up.
Switch to Incognito/In Private/Private browsing to make sure you have a clean slate.
Go to https://forums.obsidian.net
Reject the social media cookie (you can reject all others if you want, but this is the one that is causing the issue).
Log in
Try to post with quotes and multiple lines.
???
Profit (or facepalm if you're the poor sod who has to fix this )!
So, while I am not 100% certain that this is caused directly by rejecting the social media cookie - or rather, not accepting it - it is an easy way to reproduce the issues of our posters. It also neatly explains why it is only posters fom the EU that seem to be affected. There was an issue with the social media cookie settings not sticking in the past too, as @Gorgon and @HoonDing can attest to, for which there als are posts in this thread and a solution was sought.
There is one really, really, REALLY fun thing you can and should do. Once you got the steps down to reproduce it, reject all the cookies, open up Chrome's dev console by pressing CTRL + SHIFT + I, and try to post from Chrome.
You will find this will actually cause posting to behave normally, until a page refresh (going to some other page, or pressing F5). After a page refresh, it will keep eating lines and quotes until you close the development console again. Either way, the moment the scripts seem to realize that the dev console is open (or thinks it is, when you actually close it), it switches to a debug mode where the problem does not happen.
If you keep the console open and refresh, the posts get mangled again, but other than some minor warnings about deprecated jQuery functions and content policy header settings there is no error to be seen.
Fix for the posters affected is to accept the cookies. Perhaps clear out the browser cache for the forum. At least until it is fixed. Or alternatively, use Chrome and press CTRL + SHIFT + I every time you load a page, but that's certainly really annoying.
Edit: tagging in @SChin
Edit 2: Same behaviour in Edge, not much of a surprise, same basis as Chrome. Does not seem to work in Firefox from @Azdeus's testing. Can't check now, perhaps later.