Is there any device or way to measure how many watts a computer's processor is using?
Some motherboards, especially laptops, have this as part of their sensor suite: https://unix.stackexchange.com/questions/68531/return-value-of-current-watt-consumption-on-command-line. 

Also, try OpenHardwareMonitor for Windows.

The power supply for a large CPU will usually be put through a switchmode converter right next to it on the motherboard, and that itself may be a microprocessor or dedicated IC. Given that these measure their output voltage and through current as part of their operation, that is the ideal point to take a measurement. Note that it may vary a lot from one microsecond to another depending on CPU load.

The power management systems will want to know the power consumption so they can adjust the fans, the CPU speed scaling and ACPI power state in order to achieve best performance without overheating.

Edit: a device that fits in the socket is called an "interposer", and looks like this. 

Intel also sell them, although with a price listed as "build to order" that's going to be very expensive. The problem in building one is not disrupting any of the data signals.