Apple and Google achieve the same outcome (i.e. something called "card emulation", where an NFC chip can act as an emulated ISO 14443-4 smartcard), but they achieve it through very different ways:
Google just has an Android API for it called HCE (Host Card Emulation), and anybody can write an app that implements it (i.e. Google Pay has no special position compared to competitors). In a nutshell, you just get a callback for every APDU (protocol message) the phone receives from the reader and get to respond as you wish.
Apple embeds a secure element in their devices, which is a chip almost identical to that you'll find in actual physical cards, but with an additional interface that connects it to the application processor, so that the OS and (privileged, i.e. Apple Wallet only) apps can interface with it and load new card applications. That's why the storage in Apple Wallet is limited to 50-ish cards, but Google Pay allows many more :)
Felica is not part of the ISO 14443 family, but closely related and also an official physical layer of NFC (NFC-F), so many devices practically support it as well. To my knowledge, there is no software-based emulation for it though (that's always a bit risky for stored-value cards), so Suica etc. only work on Japanese phone models that have the necessary secure element, as well as on all iPhones (Apple installs a Felica applet into their secure element on demand).