http://labor.alaska.gov/lss/forms/sum-wh-act-1.pdf is the Wage and Hour Act summary, where you'll find listed exemptions:

"An independent taxicab driver who establishes the driving area and hours, who contracts on a flat rate basis for use of the cab, permit or dispatch services, and who is compensated solely by the customers served;"

Many cab companies in Fairbanks try to dictate "driving area and hours" to drivers and have refused to lease a cab to those unwilling to play along. They view the whole "independent contractor" thing as a liability / tax dodge and effectively treat their drivers as employees

As for the state, this exemption passed into law in 1993. While the taxi business did lobby for it, I know many folks who served in or worked for the legislature at the time. I was told that the Teamsters Local 959, which had fallen rather far in the post-Jesse Carr era, was trying to organize cab drivers in Anchorage and there was concern on their part about that.