Do I own the copyright automatically?

In most countries, copyright exists automatically the moment you create an original work and fix it in a tangible form (like a manuscript). You don't have to do anything special to own it.

The © notice on your copyright page doesn't create the copyright — it simply announces it, which is standard, expected publishing practice. Registration is a separate, optional step with its own benefits — see Is copyright registration required?

(This is general information, not legal advice — see Is this generator providing legal advice?)