Well, the most relevant counterexample here would be the California Health and Safety Code Sec. 50052.5 [0], which provides the cost-based definitions of "affordable housing" which apply to various income tiers (which are also those relevant to the present proposal from Gov. Brown.)
(These definitions are often prerequisites to certain subsidies, but it is meeting the definition, not the subsidy, that makes it "affordable housing".)
> The "affordable" units are turned over to a government agency
Proposals (including the present one from Gov. Brown [1]) conditioned on a certain share of affordable housing usually require that the developer make a commitment that they will meet the applicable cost-based definition of "affordable" for a period of time, and that that commitment will be enforceable by a public agency (the present proposal requires that this be done by the developer recording a land-use restriction enforceable by the public agency.)
This does not require that the property be turned over to (or even managed by) a public agency, though certainly there are affordable housing developments that have occurred where there was either direct ownership or management of the units by a public agency, nor does it require that the housing be subsidized in any way (though at the "lower income" tier, the population for whom they are targeted to be "affordable" to is also the population eligible for Section 8 housing subsidies, I believe.)
[0] http://www.leginfo.ca.gov/cgi-bin/displaycode?section=hsc&gr...
[1] http://www.dof.ca.gov/budgeting/trailer_bill_language/local_...
In my experience, as appears to be the case here, local governments often "encourage" developers to offer below-market-rate housing as part of the residential zoning rules. Here's an example: http://gothamist.com/2015/12/18/de_blasio_zoning_affordable....
When bills like this talk about "affordable" housing in new developments, they don't just mean "the rent in this unit has to be kept below $X per month" (as in rent control). The "affordable" units are turned over to a government agency, such as the Mayor's Office of Housing in San Francisco (http://sfmohcd.org/), which administers them and decides who gets to live in them. For example, last year, the city of San Francisco decided that people who lived in the same neighborhood as a building would be first in line for that building's affordable units (http://www.sfexaminer.com/affordable-housing-lottery-being-t...). The building owners' approval was not required, because the program is run by the city.