LastPass

App-specific passwords are only needed in very limited cases. See the following from Using app-specific passwords - Apple Support

App-specific passwords are passwords for your Apple ID that let you sign in to your account and securely access the information you store in iCloud from a third-party app. For example, use app-specific passwords with mail, contacts, and calendar services not provided by Apple.

Even if you did need to use an app-specific password in one or two limited cases (e.g., logging in to iCloud email from the mail application on an Android device), that wouldn't have any effect on your decision to use a password manager; i.e., you could (and I would recommend it) use a password manager whether or not you need to use any app-specific passwords.