No. Members sign in with the email and password you set up in Team and work through the owner's connection, limited to the pages and accounts you've allowed. They don't provide a token or go through Meta login.

There is one extra option: if the member has the Admin role, you can select Connect Meta (the profile goes to your account) and Connect Google (the profile goes to your account). Then they see Connect Meta or Connect Google in the menu, and any profile they connect goes into YOUR account, as if you had connected it. This lets a manager who handles the client's BM connect without needing your password. These boxes are never selected by a shortcut, and the Collaborator role doesn't have them.

If a member says they can't see any accounts, it's almost always their list of allowed accounts, not the connection.