Network error since the macOS Catalina update

It's a DNS resolver problem inside the macOS. It's not picking up the DNS servers provided by the router or inputted manually in the "network settings". The actual Internet connection is still there - you can confirm through running nslookup apple.com 8.8.8.8 through the Terminal.

The only way to solve this as of now is to reboot the Mac. Killing the mDNSResponder doesn't help.

I would also recommend contacting Apple Support through the chat, explain them this problem (and the solution). They will schedule a call with second line of support. You explain them the problem again and they will send a note to developers. I already sent my report in - more reports should expedite them releasing a fix.