Does the iPhone SE have second-generation Touch ID?

No, the iPhone SE features first-generation Touch ID components.

Only the iPhone 6s and iPhone 6s Plus are currently offered with second-generation Touch ID technology. Apple describes the newer version as up to two times faster than the original Touch ID at fingerprint recognition.

Touch ID makes it possible to use fingerprints in place of a passcode to unlock the iPhone SE, or confirm transactions with Apple Pay.