You should be able to change the behavior by going to the app Settings, and changing the Vault Timeout Action to “Log out”. Also make sure that you have disabled the option Close to tray icon.
Most Bitwarden users actually stay logged in on their apps all the time. One benefit is the convenience of not needing to supply your master password and 2FA each time that you need to access your stored credentials — instead, one can use a shorter password, a numerical PIN, or biometrics (e.g. fingerprint or face recognition) to unlock the vault whenever one wants to access vault information throughout the day. Another benefit is that if there is a temporary server outage at Bitwarden, or a temporary problem with your internet connection, you can still view your vault data using your Bitwarden app, if you haven’t logged it out.
So a typical configuration for most Bitwarden users is to set the Vault Timeout Action to “Lock” and set a short Timeout period so that the app locks when not actively being used; in addition it is common to enable Unlock with PIN (where the “PIN” can also contain letters and special characters in addition to numbers) — on browser extensions and mobile apps, the Unlock with Biometrics option is also popular.
To get the full experience, you should use the browser extension version of Bitwarden on your Mac (and the Bitwarden mobile app on your iOS devices). Unlike the browser extensions and mobile apps, the Desktop app does not have the ability to autofill login credentials on websites, which is a more secure and more convenient way of logging in (as compared to copying and pasting information from the Dekstop app).