[zokier]

What "other information" is there to verify?

[tcdent]

Hardware ID/MAC address, for example.

[xamolxix]

WLAN only cares about SSID. This is how you get roaming mode where you move between access points without losing connectivity (most Universities, company campuses, etc will have this setup)

[nandhp]

And how would you verify the MAC address of the router at the coffeeshop/train station/bus?

[lucb1e]

You can't, of course. But even if you could, many networks allow for ARP spoofing so connecting to the right access point is not really the solution. And if ARP spoofing is not possible then you run airmon-ng and wireshark.

[johnthedebs]

You would have it stored from a prior connection to the network. Of course, it's trivial to spoof that too so the additional check doesn't do any good.