It seems It used to be.
A quick Google search ends up with lots of 'vintage' and '1980s's references and pictures.
A 1989 Chicago Tribune article says this:
In the last five years, the accordion-pleated shades have popped up in cars from Valley Forge to the San Fernando Valley, from Key West to Cape Cod. They`ve become as much parts of the landscape as bumper stickers. Garfield and Odie, Batman, the California Raisins, ALF, Disney characters, women in bikinis, the Chippendales male strippers all can be yours for about $2.99. On the flip side of nearly every shade is the message, ''Need Help; Please Call Police.''
I would say that because they were so common when the director or the writer would have been growing up, that it's pretty much etched in their memories - and might well have been thought of as a nice touch to include.
Addressing other points (made in comments):
Is this a common thing in American rentals?
Not that I have seen, though it may have been part of an accessory pack. Especially in the sunshine states where the vehicle might have been exposed to the sun in a parking lot all day every day, then a cover may have been used to preserve the plastics (dashboard, steering wheel, etc). Whilst parked it can also help to keep the interior a little less hot by reflecting direct sunlight.
What is the purpose of this?
The 'help' part is a request for assistance (whether a breakdown, or more serious incident). These were common in the days before mobile cellular was available to everyone, and the distance to the next public phone might be quite far, or was unknown, so this would be the next best request assistance notice - usually in the form of a passer-by that might make note of it and call the emergency services to inform them of the notice.
Advertisers noticed the other side being blank, so this was utilized as advertising space.
So, before changes were made to vehicles (and society in the form of mobile communications) that made these less relevant, they served the purpose of keeping the dashboard and vehicle internally cool in a hot sunshine environment, a request for assistance notice, and advertising space. In the case of the movie it was also added privacy.
I caught something else:
The sun shade did not come with the rental. It was bought in the supermarket (Breaux Mart, found in New Orleans, not Florida...), along with the pack of meat, liquor, sleeping aids and other items. So, I guess to answer the question - no it is not supplied with rentals, so one needed to be bought.