Most devices will connect if the SSID and the security protocol is the same. If they set up a network with the same SSID and security protocol, using the same password (that they asked the waitress for!), then devices will automatically connect to it.
What "other information" is there to verify?
Hardware ID/MAC address, for example.
And how would you verify the MAC address of the router at the coffeeshop/train station/bus?
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.
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.
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)