Yes, it is Nat King Cole, and, yes, it is “(Get Your Kicks on) Route 66.” But there are subtle differences. I’ll let Nocaro, the person who posted this to YouTube, explain:
The Nat King Cole Trio was the first to record this Bobby Troup tune on March 15, 1946, and almost fifteen years to the day and what must have seemed a world later, Nat re-recorded “Route 66” for the benefit of stereophonic sound in 1961. “Route 66” was a hit in ’46 and will always be associated with the voice and piano of Nat King Cole.
Replicating an old song 15 years later after it was originally recorded is harder to do than it sounds. But, to my ear, Cole mostly did it.