The clutch was a part of 19th and 20th century Human automobiles.
The inhabitants of Sigma Iotia II used automobiles patterned after those found described in Chicago Mobs of the Twenties. Captain Kirk managed to operate one of those vehicles after remembering the relationship between the clutch and the gears. (TOS: "A Piece of the Action")
When B'Elanna Torres expressed an interest in learning how to drive a 1969 Mustang in 2375, Tom Paris warned her that the clutch could be tricky. (VOY: "Someone to Watch Over Me")
When Paris crashed his automobile in the Fair Haven holoprogram, he blamed it on a clutch malfunction. (VOY: "Spirit Folk")
A Grenthemen water hopper was another mode of transportation that had a clutch. The vehicle had a tendency to stall if one popped the clutch. (TNG: "Peak Performance")