@oristarA,
The term peer-to-peer is very recent. It was invented in the 1990s to describe software like Napster, where individuals could connect and share files without a central server controlling the transfer.
Peer to peer ridesharing means that two individual people, one with a a car, can connect and share a ride without a taxi-service controlling the interaction.
(Personally I don't think the metaphor is a good one... since in fact there are central companies organizing the transfer, but since "peer-to-peer" is a cool new word, some marketer thought it would be a good way to describe their product)