I see the terms watermark and digital signature being used interchangeably, but they are different:
Good watermarks are hard to remove, but might be easy to forge. Watermarks help the person who created the work to be able to prove they did it.
Fingerprints are used to trace people against their will. The person who created them didn’t create them on purpose.
Good (digital) signatures are hard to forge, but might be easy to remove. Signatures help the person who received the signature to prove agreement of the signer — typically to a contract, but also to certificates.
How is it done in the physical world?
There are two ways to sign documents. You can either write your name or you can use a stamp. In former times, a seal was created. All ways are extremely insecure as you essentially only need to see the original to be able to forge a physical signature / stamp / seal.
Creating watermarks is more time intensive. I also don’t know any way to remove a watermark from a physical document.
Fingerprints are known to everybody, but did you know that your printer might add them as well? It’s called machine identification code or tracking dots. Those are tiny yellow dots that some printers leave on the page in order to make it identifiable.
How is it done in the digital world?
Digital signatures use public key cryptography like the RSA cryptosystem and ECDSA. If implemented right, state-of-the-art digital signatures are impossible to forge. It’s hard to explain how they work, but Computerphile does a good job:
Watermarks are trickier. How do you ensure that somebody does not remove the information that you created some work from the work?
The simplest way to add a watermark to a file is by changing metadata. However, that is typically easy to remove. For pure images / audio / video, the user could simply take a screenshot / redirect the audio output / record the screen.
A solution for images that worked for a while was to add an overlay over a key part of the image. However, with recent AI improvements that doesn’t help so well anymore.
For audio, you could also add beats in a certain pattern. The key part is that they must be clear to hear.
Fingerprints also exist for devices. You can identify browsers by a lot more than just the user agent, see: