You’re all wrapped up around the differences in the network path.
A web site taking different actions based on source address is not at all unusual, though prolly more commonly encountered involving source IP address access control; allowing and denying access for ranges of IP addresses. Which is where we get folks using VPNs to work around those same blocks, for media content providers.
For this case, why wouldn’t a web site want to provide (slightly) different content in a broadband path and a mobile broadband path? Content and images for mobile, in this case. The web site might also have a deal with a content delivery network, which is another case where your IP address can come into play; trying to keep the content physically located closer within the internet if not within the carrier or ISP network, and provide content tuned for traversing a mobile carrier network.
And then there’s the utter shambles of online advertising and tracking (and sometimes content injection), and who really knows what dreck arises from that.
What many folks think of as a single web server causes your browser to assemble and display a web page within your browser retrieving that data from dozens or hundreds of different and geographically diverse web servers. Then we get into content delivery networks, DDoS protection, spam filtering, DNS, and other fun topics...
Neither the web and IP routing are much like two-cans-and-a-string. Not by a long shot. Tends and hundreds of cans and strings, maybe... Even for what might initially seem like a two-cans-and-string connection...
