I don’t use iMessages but it is clearly running in the background some place. (The computer interrupted a movie to ask me if I wanted to add my phone number. No!) Is there any way to make Messages shut off entirely?
[Note my newest computer is running Sonoma 14.5. I also have some much older units running 10.13.x along with an old iPhone and an up to date iPad Mini.]
I’m not surprised that you were auto-logged-in as a part of logging in to your iCloud account, but you can log off.
On Sonoma, launch Messages. Then go to Settings → iMessage and click “Sign Out”. You will be logged off from iMessage on your Mac.
If you later want back in, just launch Messages and log-in using your Apple ID credentials. After logging in, all your other devices should present a notification informing you about the login (which would indicate a successful break-in if you weren’t the one logging in).
Older versions of macOS should be similar. Log in/out via the Messages app. The specific location of the command will vary, but it should be somewhere within the app.
Done. Thank You.