There's this specific bot that I see posting on Rule 34 ( https://rule34.xxx/index.php?page=post&s=list&tags=user:bot ) and I'm noticing that some of its posts are copied from e621 posts. "copied" as in, they post the same art and use the exact same tags. Completelt identical tags. ThatDawgMurray's posts specifically are some where this is notable, https://rule34.xxx/index.php?page=post&s=list&tags=thatdawgmurray+user%3abot

Is this a deliberate collaboration between the two websites? Also, does this repost everything, or do I need to activate a certain setting on my account for my e621 posts to be mirrored by the bot? It would save me the time of posting on both websites.

In general, I am curious to learn more about this bot

On e621's side, it's certainly not an intentional collaboration - e621 does not block such bots, but it doesn't run any of the cross-posting bots either.
That said, there certainly are bot posts from e621 to rule34, it's just not anything official.

topic #55270
topic #41914
And I'm sure there are more

These things happen when apis are easily accessible and there's really nothing anyone can do about it

From my own observation of what it's taken from my own posts, it seems to only take stuff that's explicit, tho stuff tagged with 'sketch' didn't get taken.

Basically, the Rule 34 bot's a hostile parasite that we can't figure out how to remove without cutting off our own arm.

They are definitely not our collaborateurs, even if quite a few artists have accused us of using it as an excuse to keep DNP artwork posted. (We don't, to make that clear.)

Any public art site can run into a similar problem. As long as it's possible for users to see art posted by other users, there's nothing preventing a third party from making a bot to automatically repost it to some booru. e6 just happens to be a juicy target because it's fairly well known, curated for quality, and has a lot of effort put into tagging for searchability and blacklisting reasons, so other sites want to take advantage of that work. Notably though, e6 itself does not employ such bots to repost from elsewhere or to elsewhere, and accounts found to be using bots to post here get banned (if for no other reason than because automated tagging, or tags copied from other sites, tends to be poor).