How can I whitelist applications for users? We had few MacBooks where users without admin rights had Parental Control turned on and could open only whitelisted apps. It was working very well and it was everything we needed really.

With macOS 10.15, Catalina, update Parental Control is gone and all we have now is Time Screen which doesn't do anything like Parental Control. All I can do is time limit to (not every) application and I can't whitelist any applications, or set time limit to 0:00. 

Is there any other solution to restrict user to use only whitelisted apps?
Under Screen Time there's a setting called "Downtime", it's for specifically for white-listing apps and you could schedule it by day. Any new apps introduced will be blacklisted by default, at least that's the way it works on iOS.