Is there a difference between an NFC and RFID chip?
NFC is a type of RFID. Not all RFID devices use NFC.
NFC is a bundle of specific technologies and protocols used to communicate with storage devices (often credit cards or passive identifier tags). The NFC specifications cover everything from the radio frequency used (13.56 MHz) and the types of modulation used, to the communications protocol used to read and write data on tags.
RFID, by contrast, is a generic term used to refer to any form of radio communications with an identifier tag. There are dozens, if not hundreds, of different systems which fall under this category, and most of them are mutually incompatible. NFC is often considered a type of RFID, but they are not synonymous.
Pet microchips do not use NFC. There are a variety of different RFID systems used for these chips; most of them operate at 125, 128, or 134.2 kHz, and they are largely incompatible with each other. (It is not unheard of for an animal to be implanted with multiple brands of ID microchips to increase the odds that they will be identified!) A NFC reader in a cell phone will have no way of communicating with these chips.
Your friend is talking through his hat, they're related but not the same.
If you read the wikipedia page about NFC you can read there that:
NFC is a set of short-range wireless technologies, typically requiring a separation of 10 cm or less. NFC operates at 13.56 MHz
Remember that value of 13.56 MHz
Then read the wikipedia page about RFID and scroll down to the table under Frequencies
Note how several frequency band are listed, 120 - 150 kHz, 13.56 MHz up to 10 GHz.
Also note how in the 13.56 MHz row "NXP's Mifare" is listed, which is an NXP product using NFC.
So RFID is much broader.
All NFC product could be called RFID (as they fall in the 13.56 MHz category).
Not all RFID products are NFC, the RFID products that do not work on 13.56 MHz are not NFC.
Allow me his silly comparison: Ducks (NFC) are birds (RFID) but not all birds (RFID) are Ducks (NFC).