Does "Natel" mean the same thing as "Handy" (cell phone) in German? Why and how?
Yes, it is used that way in Switzerland. Note that you will most likely not be understood if you use the word outside of Switzerland. It is a Swiss Standard German word (cf. Duden | Natel).
The origin is the Swiss brand NATEL ("Nationales Autotelefon") of Swisscom (or originally of the Swiss Mail & Telecommunications service PTT).
Wikipedia has more.