Apps reopen at login/restart despite having selected the "close app when quitting"

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Some Applications Close when the Windows of that application is closed but suggest looking at the Dock to make sure the small dot below the Application id Not Appearing.

The small dot indicates the application is closed but Still Loaded and active to be reused without having to actually Relaunch it

To illustrate this I have Notes and Preview both with Windows Closed but that are Still Active and ready to open and use again without having to ReLaunch them.

Refer first image

Then there are the Applications enabled to Launch on a Cold Boot or Restart

Second image

Lastly is the check mark Enabled when the computer is Shutdown ?

Any of the applications that were Open and / or still Active as per First Image would launch on the next startup or restart

I've seen this for all versions of Mac OS X. However, for me this behaviour is somewhat random. Sometimes a reboot will return a fresh startup and other times all open apps re-appear. I must say that the reloading of open apps is less frequent for me since Ventura.

I do find that the open Finder windows always re-appear on every reboot.

I can confirm that my open Finder windows always re-appear no matter the setting of "Re-open windows when logging back in " is. And this behaviour can be reproduced when restarting in Safe mode or normal mode every time for me. And I'm running the latest Ventura version.

Also. BTW, I use multiple desktops (not monitors, but desktops) and the Finder windows that re-appear on restart also always disappear on the desktops other than Desktop 1. I've had an open call on this many times, but it also seems to disappear into the ether. I am told it's being worked on and I'll be contacted and when I check I find my call is closed, with no further communications from Apple. And this has happened at least 2 times now.

You are welcome and thank you too