The old call screening option had an option nested in it, that would optionally ask callers for their name the first time they called you, and then never again. So, you had three choices: don't use call screening at all, use it, or use it AND ask for the callers' names.
The new call screening option works much differently. There is no user-accessible setting to turn the "ask for name" thing on or off. If call screening is turned on, it will ask all callers whose number(s) are not in your Google Contacts to state their name, every time they call, until you add them to Google Contacts. At that point, it won't screen those callers at all anymore.