Netgear GS 308E, throwing Unidentified Network when using laptop

Everything is possible, it just requires some basic or essential networking and VLAN knowledge.

This problem starts with the fact that most ISP modem issue just one public routeable IP address, which is of course taken  by the connected GS308E (resp assigned by DHCP to the GS308E). And NO: It's not the GS308E assigning a ZeroConf LAN IP address to your computer: The computer does it on its own if it does not get a workable LAN IP address.

Of course, you could configure the GS308E to a management LAN IP address in a freely chosen IP subnet, ideally (for simplicity) one where a router does manage and assign LAN IP addresses in a NATed way. Anything else is really hardcore and not user friendly. Sure, you could put the switch LAN IP to a private LAN IP subnet like 192.168.123.0/24, setting it for example to an address like 192.168.123.10 255.255.255.0 gw 192.168.123.1  ... in absence of a true management VLAN on the GS308E (what makes it much simpler), you can temporary put a computer network adapter adapter another address like 192.168.123.20/24 and again mask 255.255.255.0 gw 192.168.123.1 to reach the switch management Web UI if desired.

Last but not least, what for - and without any networking knowledge - do you think you need a switch connected to the modem, and then .... ?