Cisco anyconnect vpn does not allow local LAN access on MACOS Ventura after reconnection

What you are trying to use is called "split tunnel." Most VPNs use "full tunnel." Split tunnel separates the traffic into the VPN path and a non-VPN path. The implementation of split tunneling can be problematic, that is, it partially works under some configurations. My employer requires VPN for remote access, and has switched from Cisco Any Connect to a different product; we also can no longer use split tunnel because it has security vulnerabilities. This is inconvenient: for instance, without split tunnel, I cannot print on a local network printer without first disconnecting from VPN. Since Ventura makes some changes to security and networking interfaces, that indeed could be why this happened to you. Or there may be something unique in your network configuration that is incompatible with split tunneling. If you need the local LAN while using VPN, you may need to switch to a different VPN.

VPNs are controversial. Employers and banks provide customized implementations of VPN that enhance security, significantly. However individuals just using a commercial VPN provider may be degrading their security without knowing it. Some of these VPN providers actually store and sell your private information that is lifted from the internet traffic that you thought was secure. There have been scandals and lawsuits about this. And in some instances, you might have no way to ascertain if this is taking place.

Unless you have a specific need for a VPN, say by your school or employer, you should discontinue its use because of problems exactly like what you are experiencing.