Anthropic judge rejects $1.5B AI copyright settlement

Remains to be seen, but generally the holder of copyright is the author not the publisher.

That depends on the publishers and your standing with them. Many publishers want a copyright transfer agreement whereas others are fine with exclusive licensing rights. You can't transfer copyright in some countries (e.g. Germany) but you can in the US.

Even though the US allows copyright assignment, none of the Big Five publishing houses in the US require it as part of a standard book deal, even with first-time authors. If you open any book or ebook to the copyright page, unless it's something like a reference book (which are frequently work-for-hire), it will say some variant of "© Author's Name."

Publishers get exclusive print publishing rights for a given market, typically get digital and audio publication rights for the same, and frequently get a handful of other rights like the ability to license it for publication in other markets. But ownership of the work is almost always retained by the author.

I don't think you should work with a publisher who wants a copyright transfer. It is not part of standard book deals.

Even exclusive licensing rights very often have limitations to them such as a duration or requirements to keep the license, and people should be vary about working with a publisher who wants exclusive licensing without termination clauses that protects them as well.