My broker shows Implied Volatility for a stock, but as I understand the BS model, it's calculated on a per option basis. So how is it shown for the entire stock?
What you are referring to is the Composite Volatility which is an evaluation of all of the options of a single security.
There are a variety of ways to calculate it. One well known author/service weights each individual option's implied volatility by its trading volume and its distance in or out-of-the-money. Another popular service calculates it by weighting delta and vega of each option.
The Composite Volatility number may vary somewhat from one method of calculation to another. That's not critical. The importance of the calculation is that the daily numbers can be assembled and volatility charts can be displayed in order to see how past daily implied volatilities compare to today's Composite Volatility.
IVolatility is one of several that offer this (free sign up).
You are correct - there is not a single implied vol for a stock. You'd have to ask your broker to be certain, but most likely they are quoting the at-the-money (ATM) vol (implied vol for the put or call with the strike closest to the current underlying price) for the prompt option contract (the contract set to expire next). It's possible they could be taking an average of the ATM Put and ATM call vols, but those should be very close together.