How do you know if your IP address is being monitored?

You have to be a little bit more specific. Are you saying that you're using a computer in a public library, and your computer is logging the IP address of that computer? If so, then no, the IP address isn't being monitored. It's just a machine with a public IP address. If you're asking "is the library spying on me" then I'd say that's a whole different question.

The only way to determine whether or not your IP address is being monitored is to check your IP address log in a public library. If you don't know how to do that, I suggest that you either ask a librarian or contact your local school.

Why would a device have multiple IP addresses?

From a networking standpoint, I've never heard of a device having multiple IP addresses.

If I have an IP address for my home network, and I plug that same device into my office router, would the device have two IP addresses (one for each router)? The device could have multiple IP addresses for a number of reasons. The device has two or more network interfaces, and is trying to connect to both at the same time. This can be the case when a device has more than one network adapter - either because the manufacturer/firmware/driver decided to support both, or because they have a USB interface on their board.

Multiple IP addresses could be necessary if the device is in a subnet with multiple IP ranges. For example, if your device was located in a small office with a /24 network range and the office router had a /24 network range, your device could have a single IP address in the local subnet, but multiple IP addresses in the router subnet.

If the device has multiple IP addresses, it's not the same as having two devices.