Is frame number the same thing as serial number? I'm registering my bike with Kryptonite anti-theft scheme and it asks for the serial number. My local dealer wrote down the frame number for me, saying I might need this for insurance purposes. Are these two names for the same number or are they distinct?
Yes, two phrases for the same thing.

More specifically its the frame's serial number.  Other components may have their own serial numbers too.

Any electric shifting component will likely have serial numbers which can be read by software, as will power meters or other sensors.

Small priced items generally don't have serial numbers.