First off, the mechanism is DHCP. Now, the remaining questions are
How does assignment work?
Did I get a public IP address?
Is it static?
Assignment is pretty much up to the ISP. A strict ISP is going to be maniacal about tracking every single piece of equipment on the network, maintaining a database about all subscribers and MAC addresses and then doing an assignment for that subscriber from the appropriate prefix.
Most ISPs are still using public IP addresses, but this is not a requirement. It would be very easy for an ISP to put all of their subscribers behind a carrier-grade NAT system. I’m pretty sure there are a few folks already doing this. So, your IP address may be out of RFC 1918 space and thus not usable externally.
ISPs seem very loathe to assign a static IP address. If nothing else, they charge more for them. However, they don’t seem to commonly let addresses float either. An ISP that was desperate for address space could easily reuse addresses when subscribers are not active.