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.
Frame numbers are generally required to be stamped on every frame. They are usually 6-10 digits long.
However, a very small minority of bikes (notably Bromptoms) also have a barcoded serial number that is affixed on a sticker.
In this photo you can see both the serial number (15033303xx) and the frame number (49374x). The serial number embeds a date code, so you can tell it was assembled in March, 2015. This helps the manufacturer track defects in assembly as well as sales data for warranty verification. When you register your Brompton, you supply both.
Generally police and bike registries want the frame number (even if they ask for the serial number) as the serial number sticker can be peeled off. If you register your bike on BikeIndex.org (strongly recommended), then you can include both.