It uses the IP address of the computer... it does not have it's own IP address.
That's not really true.
Computers don't have IP addresses; only an OS running on a device will have an IP address. If you run different OSs on the same computer there is no necessity - or guarantee - that they will use the same IP address. (An obvious example is if you use fixed addresses; there's nothing to stop you using different addresses for different OSs. And if both use DHCP they may at different times acquire a different address for the same OS, let alone other ones.)
In answer to the OP, live USBs almost invariably use DHCP addresses; they may, or may not, be the same address that - for example - Windows acquires. Bottom line - it really doesn't matter.
Rider - all of this refers to the internal LAN IP address. As far as the outside world is concerned all computers on your LAN use the same IP address; but, again, this may well vary over time.