[Long-time 10k+ MO user here, asking this question anonymously for reasons that should be obvious. Added Oct 16: My access to this account will self-destruct in 3, 2, 1 .... (sorry, won't be able to accept an official answer. For what it's worth, I strongly support Tim Campion's suggestion.)]
This question is not meant to be a place to discuss the pros and cons of the dumpster fire on meta.SE, or the underlying issues that started it. Go there if you want to get depressed. Also, I think that only moderators are in a position to answer this question.
The short background is that StackExchange Inc. 1) is trying to implement a new Code of Conduct (CoC), and it's proving difficult, and 2) has retroactively relicensed all existing content, questioned by many as being of dubious legality. These are separate issues, but have both contributed to a wide feeling (expressed through community voting and extensive commenting) that SE Inc. is making a mess of things, one way or another.
Since MathOverflow uses the SE platform, but is actually a different organisation it is not clear what exposure (in the sense of risk) MO has to the recent controversies. Are our moderators bound by the same agreement as the SE network moderators? Will the CoC apply here, when it is finally sorted out? Were contributions to MO retroactively relicensed (making this question outdated)?
Again, the intent of this question is not the content of the CoC or the relicensing issue, but to get in writing the facts of where MO stands in relation to these from a contractual/legal/etc way.
Edit (To clarify some things) I asked this question in a slightly vague way, partly because I didn't want to pre-empt any position or bring up debates here. But I've read a bit more, and Johannes pointed out some relevant facts in an answer. So here are some more focussed questions. Again, this is not to debate the merit or otherwise of the below points, I'm looking for official answers from mods or MO owners/board members on policy about changes that StackExchange Inc makes.
StackExchange relicensed all contributions users had made to their websites (from CC-By-SA 3.0 to CC-By-SA 4.0, which change the CC people say can't be made without permission), but this has also relicensed all MathOverflow content too. Is this compatible with the agreement between the MathOverflow corporation and StackExchange Inc?
I presume that there is very little objection to the principles of the new Code of Conduct (CoC), and find it nigh-impossible to forsee any circumstances in which MathOverflow as an organisation would object to the now in-position wording, and adopting the CoC as official here (were it a choice). However, the official FAQ on the CoC, which has given rise to concerns among many StackExchange users, is getting rather distant from any document that I can see referred to under the legal agreement between MathOverflow and StackExchange Inc. The CoC is linked to from MathOverflow pages, and so is, it seems, in effect and binding. I haven't seen, after some cursory clicking about, explicit links from MathOverflow to the contentious FAQ. Does this latter nonetheless carry force on MathOverflow to the extent it does on the StackExchange network? Will all future policy changes in the same manner also immediately and automatically apply on MathOverflow? (Added: I note that this type of question was raised nearly six years ago, to which it seems the official answer was, to paraphrase, MathOverflow has an informal understanding with StackExchange Inc that SE Inc employees shouldn't interfere with our moderation as we can choose to leave the network. However, this seems to me to be a bit like the nuclear deterrent...)
Perhaps all of this is moot. But I hope MathOverflow is not hit with the same
ugly stick terrible optics that StackExchange Inc has encountered in trying to create an environment that once was expressed through 'be nice'.
Update: Yesterday a new FAQ regarding the pronoun discussion has been released. This new FAQ is based on previous input from the community and it seems that the acceptance of these new rules is (so far) much higher.