If I plug in two separate strings of christmas lights into each other, is the voltage multipled by number of parallel circuits?
No - each set of lights still sees 120 volts, and the whole assembly still needs 120 volts.

The 120 volt line and neutral wires from the plug on the first set go directly to the socket on that set.  The strings of lights in that set are wired between the line and neutral wires, and do not interfere with the 120 V connection between plug and socket.

As far as the second set is concerned, the first set is just an extension cord.