You are quite correct, it's the MAC address of the router and it's for this exact reason that most home routers let you change the MAC address (As some ISP's are particularly fussy about a MAC address changing).
I'll be honest, I'm not 100% certain how it works in a PPoE environment because in the UK at least, very few ISP's actually use PPoE and it's generally possible to change your equipment whenever you want. Our big Cable ISP is the exception to this and you HAVE to use their equipment, but you can just treat it as a dumb modem and the router just connects via Ethernet and the DHCP requests are forwarded on.
In any case, while I agree that the OP is lacking in some basic networking knowledge, I do think they're likely correct in what they're saying about MAC changing causing an IP change.