This is certainly possible if your ISP is using CNAT.
The most common CNAT IP range starts with 100.X.X.X (you can find it on the WAN port of the router).
CNAT is an attempt to preserve constantly shrinking number of the public IPv4 addresses.
The idea is similar to standard NAT (Network Address Translation) with an exception that it does not only use ports to route the traffic to your computer but it also uses your "public" (not really public) CNAT address instead of private IP as it is in case of a traditional NAT.
The benefit of this solution (in addition to saving real public IPv4 addresses) is that it's much cheaper for the customer and provider.
For example your ISP is having a RIPE range of 100 public IPs and behind each of them runs a CNAT subnet of 100.0.0.0/8
That gives them over 16.7 million CNAT IPs per single public IP.
In this case you and your friend that is using same ISP will see same IP address via What's My IP website (which is the IP of CNAT gateway router of your ISP) and different CNAT "public" IPs on the WAN port of your router.
The drawback of such solution is that you cannot really publish any service to the public internet, although it may work for the clients that are sharing the same CNAT subnet and ISP with you.
It's basically like a fake public network inside real public network.
I hope it makes sense and helps.
