Ok follow up - I tried it but it didn't work.
Somehow it shows up as connected under wlan0 with ifconfig (shows connected ssid of the network) but it will only let me ping 8.8.8.8 - no other site works like even www.google.com
If you can ping IP addresses but not by name, you're not getting DNS resolution from your upstream, presumably they'll allow DNS when you are logged in. It's a cheap way for them to deny unauthorised access.
On the mobile phone it automatically redirects to an additional verification page (which asks for phone or email) - but on the laptop it just doesn't show any such redirection when attempting to open Google or Bing or whatever.
Yeah we have a network here that works like that. Here at least, once you "enrol" it will remember your bssid for some hours or days before doing so again.
Do I need to do something to get to their local authentication/verification page?
It may not be a redirection but a pop-up window?
You could try allowing pop-ups on your browser; maybe your phone browser allowed these? No warranty ...
If not, maybe try with links or lynx text-mode browser to get logged in, or examine page source to see why not.