Actually, IP addresses are not quite unique. Internal IP addresses are unique to the network in which they are using, but even that can change. Truth is, your internal IP address can be the same as several other people's addresses, but due to NAT (Network Address Translation) they appear to an outside address, which may (or may not be) unique.
Outside addresses, which can be shared internally through Network Address Translation, can be re-assigned if you are not using a static IP address.
Modems which connect to internet services providers, which can be assigned IP addressed dynamically, may not ensure that your machine has an outside IP address which is the same each time.
If you have a 'static' IP address (which doesn't change) through your Internet Service Provider, then it will be unique to you (unless someone else creates their own 'internet') otherwise, your IP will not be unique.
For an internal network, it is most likely unique, unless you change configuration options for networking to ensure it isn't.