Is the word 'driver' is a shortened form?

No. It neither has a expansion nor a shortened form.

It is just like the following,

cool => cooler (as in water cooler)

killer=> killer

murder => murderer

build => builder (the one who builds the buildings)

drop => dropper

skip => skipper

dance => dancer

"Driver" - a person who drives a vehicle on the road.

Sometimes seen figuratively, as in "Greed is a driver of ambition", and similar.

The verb is 'to drive' You can see 'bus driver' or lorry driver' or 'truck driver' and similar as two-word expressions.

It is from the verb to drive - and it is not a shotened form of anything.

There are variants as phrases - eg lorry driver, 'bus driver, etc.