Cars are subsidized via road building/maintenance, after all, and it's likely to cost less to maintain a few bikes than it does to maintain roads/parking for the cars that would displace them.
Imagine if there were no transportation subsidies and road maintenance were privatized. A nightmare. It seems entirely reasonable that forms of transportation that can help bring tax revenues to a city are subsidized, and therefore entirely reasonable to create a start-up based in part on those subsidies.
Isn't HN usually full of people saying you have to figure out what customers will pay for (meaning, what customers will pay more for than it costs you to provide)? Bike-sharing is not one of those things. End-user customers will not pay you more than it costs to provide.
A lot of successful start-ups don't sell to consumers at all. And kazevedo comments that they are in fact structured as a B2B start-up. [1]
For example, DC's capital bikeshare comes from Federal grants that covered the setup costs, while Velib in Paris is entirely operated by JCD, an advertising agency that operates the system in exchange for advertising rights.
Take Velib. Last time I checked, JCD had revenue of $54 million, with an operating expense of roughly $35 million on Velib. The theft and vandalism complaints were regrettably exaggerated when the City and JCD were re-negotiating their contract last year. That they need subsidies to exist is patently false.
Financial aspects of US systems are also healthy as pointed out by other commenters.
On a related note, the subsidies to other modes, mainly rail and roads, are several orders of magnitude higher.