I can't see any logout option in the Stadia App, but if you are still logged in on stadia.google.com, simply click your avatar in the top right corner and choose "Sign out".
For the Stadia app I only see the option to switch your account, which would only work if you have a second account you can switch to. If not, it might help to deinstall the App and install it again as it then should ask you again with which account you want to log in. This should kick out any users which are still connected with your account on your device.
If you want more security for your Google Account, you can also protect it with the 2-step verification. In this article you can find more information about it and steps how to do so. Additionally these steps here might also help you to make your account more secure.