I've made a server (python, twisted) for my online game. Started with TCP, then later added constant updates with UDP (saw a big speed improvement). But now, I need to connect each UDP socket client with each TCP client. 
I'm doing this by having each client first connect to the TCP server, and getting a unique ID. Then the client sends this ID to the UDP server, connecting it also. I then have a main list of TCP clients (ordered by the unique ID).  
My goal is to be able to send messages to the same client over both TCP and UDP.
What is the best way to link a UDP and TCP socket to the same client?
Can I just take the IP address of a new TCP client, and send them data over UDP to that IP?  Or is it necessary for the client to connect twice, once for TCP and once for UDP (by sending a 'connect' message)?
Finally, if anyone with knowledge of TCP/UDP could tell me (i'm new!), will the same client have the same IP address when connecting over UDP vs TCP (from the same machine)? (I need to know this, to secure my server, but I don't want to accidentally block some fair users)
First of all when you send data with TCP or UDP you have to give the port.
If your client connect with TCP and after your server send a response with UDP the packet will be reject by the client.
Why? Because you have to register a port for connection and you can not be sure the port is correctly open on the client.
So when you begin a connection in TCP the client open a port to send data and receive the response. You have to make the same with UDP. When client begin all communication with server you can be sure all the necessary  port are open.
Don't forget to send data on the port which the connection was open.

Can I just take the IP address of a new TCP client, and send them data over UDP to that IP? Or is it necessary for the client to connect twice, once for TCP and once for UDP (by sending a 'connect' message)?

Why you don't want create 2 connections?
You have to use UDP for movement for example. because if you create an FPS you can send the player's position every 50ms so it's really important to use UDP.
It's not just a question of better connection. If you want to have a really good connection between client and server you need to use Async connection and use STREAM. But if you use stream you'r TCP socket do not signal the end of a socket but you have a better transmition. So you have to write something to show the packet end (for example <EOF>).
But you have a problem with this. Every socket you receive you have to analyze the data and split over the <EOF>. It can take a lot a processor.
With UDP the packet always have a end signal. But you need to implement a security check.
UDP and TCP always use same IP for one client?