Sure. Easy. Sony has their own proprietary protocol. When the PS3 comes online, it sends a message to Sony that says "My MAC is blah. Am I banned?" And Sony will reply back and say if it is or isn't. And if you do something on your PS3 to get banned, it will send a message to Sony saying "My MAC is blah. Ban me for I have done something stupid"
That is all in the data portion of the packet, what is commonly referred to as a payload. You can put whatever the hell you want in a payload packet and it won't be changed en route to the destination (hopefully, if it is, there's a huge security breach somewhere, but that's another subject).
HTTP, the protocol used for web browsing, and TBH, just about every other protocol out there, does not include MAC addresses in the payload. There are sections in the packet for source and destination MACs, and those are what's changed each time it goes through a router.
Here's basically what a packet looks like visualized:
The only part of the packet that is untouched is what's inside the blue box (and the green box, encapsulated in the IP segment of the packet). The stuff outside the blue box is changed each hop.
When you first send a request, the source MAC is your local computer, and the destination MAC is your router. Your router will then replace the source with its MAC and the destination of the MAC with your modem (or not, if you have an all in one). Then your modem's MAC becomes the source, and your ISP's router becomes the destination. Basically, each router puts its MAC in the source and the next router's MAC in the destination as it makes its way across the internet. By the time it reaches the destination server, the packet's source MAC is their local router.
Sony sends the PS3's MAC address inside the data segment since it's a specific message querying if a MAC is banned. HTTP does not send the MAC along with the request.