What version of AnyConnect are you running? You need to be running 4.8 when upgraded to Catalina as this is the only truely supported version for 10.15.
I suggest upgrading AnyConnect to the newest version and then test. Please let us know the results.
As I mentioned in my earlier post, only 4.8 is currently fully supported on macOS 10.15 Catalina.
Also have this issue with 4.8.00175:
v.4.8.01090 is the same but also v.4.7 and all three versions are the same issue under MacOS Mojave against our company ASA Server, so this might not be just a Catalina issue... Latest version I got access to that works under Mojave is 4.1
Hope you have been ale to resolve. I just found this solution and it worked for me. HUGE thank you to the University of Waterloo for sharing this. I am logged into my Remote Desktop now. I was starting to fret I had a $4000 paperweight on my hands.
Some Mac OSX users who are fully up to date with the latest version of Mac OSX and the latest version of Cisco AnyConnect VPN are still having issues trying to make a VPN connection from their home network.
When attempting to launch Cisco AnyConnect they will see a prompt stating “/!\ The VPN client was unable to successfully verify the IP forwarding table modifications. A VPN connection will not be established.”
Disabling IPv6 for Wi-Fi has fixed this issue for several people. Note, IPv6 is not used for most of North America so it can be turned off for Wi-Fi on the MacBook unless traveling outside of North America.