Much like the tags in an image, your watchlist is saved in your settings as a bunch of tags, and that bunch can be edited in what we call a Plain Editor way, which allows for easy copy-pasting. If you got to https://manebooru.art/settings/edit in your own account you can see the section I'm talking about, in the first tab titled Watch List. You can simply go to the same menu on the other site, click the Plain Editor button there, copy the list, come back to you settings here, click Plain Editor and paste it here. After that you just save your settings.
I hope that was clear, if not here's a couple images that might help. It is possible for an artist to have different tags across sites, but in my experience this is fairly uncommon. In cases like that you can ask around and maybe one of us (or a user) will know the appropriate tag.