For example if I had a gameplay server and a voice chat server, could the client be connected to two servers at once? So that I send input to the gameplay server and microphone data to the voice server.
yes 
Definitely - I have this running currently with a world-server that deals with global chat channels, and character control; and one of many zone-servers that runs a zone simulation.
I got it wrong to start with by having 2 separate clients, but that didn’t work. Now I have a single client core with 2 hosts. When messages are received, I check the host, and fire the message off to different routines based on their host.
How do you set up this? On the client side