In the U.S., bike racks are metal structures where you can lock your own bike. It's not at all like a consigne.
Other self-storage options (such as at Subway stations) can be called bike parking or bike storage or bike lockers. All of the facilities like this that I've seen (including the one with Bart in San Francisco) are for people who regularly commute from the same place ie. a subway station. I wanted to leave my bicycle there, but couldn't without signing up in advance and signing a contract! The "Bicycle station" link you provided appears to be a brand name, and also works with a long-term contract.
I've only once seen in the U.S. a bike "consigne" where any random person can walk up with a bicycle and leave it for a set period of time. This was done during a festival after a 50-mile bike ride in Boston, and it was called a bike check.