Casual Car Giveaway

"He tosses his keys to a random old black guy sitting on the sidewalk. The man is outraged, and in Wolfman Jack's voice yells, 'What do I look like, a valet?' How dare you, sir? How dare you mistake me, a guy with nothing better to do than sit on the sidewalk in the middle of the night, for a lowly valet? But there's no racial stereotyping going on here. Dalton tells the man, 'Keep it. It's yours.' The man looks at the keys, and with a brief grunt heads over to his new ride. And we all know you don't need anything else to assume ownership of a car, like, say, the car's title or any other sort of paperwork. That's Dalton for you; He just passes along his worldly possessions like it's his katra."

—The Agony Booth's recap of Road House

In fiction, if characters no longer need their car, bike or any other vehicle, they can just hand over the keys to the first guy they see. No need to worry about insurance, registration or any documentation whatsoever. You can just hand over your old vehicle and go on with your business.

Contrast Hero Stole My Bike, where the hero needs a vehicle, and thus takes it from the first guy he sees. See also Undead Tax Exemption, for a different "no paperwork needed" trope.

Examples of Casual Car Giveaway include:

Film

In Road House, when Dalton is leaving, he gives the keys to his old car to a nearby bum, as humorously described by The Agony Booth in the page quote.
In Star Trek 2009, when Kirk leaves for Starfleet Academy, he gives his motorbike to one of the construction workers at the dock:

Construction Worker: Nice bike.
Kirk: (tossing the keys) It's yours.

The Studebaker in The Muppet Movie gets traded in with relative ease.
In Tommy Boy, when Tommy and Richard go to the airport, they drive up to the main entrance, where a guard tells them they aren't allowed to park there, prompting Richard to say "Uh, keep it!" (the car was basically falling apart at this point).

Literature

In the end of the first Molly Moon book, Molly does give away her car to a somewhat randomly chosen man. However, she also gives away the documentation of the ownership, telling the man to simply sign the papers, and then the car will be his.

Live-Action TV

On Seinfeld, Jerry gets rid of a car infected with unbelievable B.O. by driving it to a bad neighborhood and leaving it with the keys in front of a shady-looking guy. The guy steals it immediately... and regrets it just as quickly.
In an episode of Life, Charlie decides that his luxury car is too obvious (there are criminals looking for him) so he offers to exchange cars with someone driving an old beater who stops next to him at the traffic lights. Once the other guy realises Charlie is serious, he happily makes the trade.
The Hustle crew do it in "Big Daddy Calling", swapping their Cadillac for an old beat-up pickup truck as they flee Las Vegas.
In the Top Gear American special, when the guys arrive in New Orleans, a year after Katrina, they were supposed to sell their cars for whatever they could get for them. They instead decide to give the cars away, after seeing the condition the city is in. Subverted in that they couldn't find anyone who would take the cars.

Music

Happens in the music video of Stratovarius' "Hunting High and Low", where the protagonist gives his car to a pair of hobos, as part of spiritual liberation from his former life.

Real Life

In 2004, Oprah gave away cars to members of her studio audience. She did not pay the taxes on them, as befits this trope, and since this is real life the recipients were stuck with huge tax bills. In a later giveaway in 2010 she promised to pay the taxes, though there were questions about how accurate the amount was.