It really helps if you would provide the exact version of macOS you are using since macOS 13.x Ventura has changed the location of may configuration options.
Check to make sure the app is not configured in the Login Items. With macOS Ventura, this may be located in the "General" Settings (I think), or in the "User & Groups" System Preferences for all previous versions of macOS.
When you restart/shutdown/logout of macOS, make sure to uncheck the "Open windows when logging in" option as well.
If the app is still launching automatically, then check the app's own settings/preferences for an automatic launch/start option.
Edit: What app is it? Some third party apps may include a driver which may launch automatically at startup regardless of any settings since the app would be useless without the driver. Booting into Safe Mode will prevent third party software (including drivers) from launching automatically during boot & login.