You're asking whether bounty as used on Stack Exchange reflects the 'normal' usage of the word in English. Let's look at a dictionary for a scholarly definition, a wikipedia reference for popular usage, and Stack Exchange for its particular usage.
1 A sum paid for killing or capturing a person or animal: there was an increased bounty on his head
A bounty (from Latin bonitās, goodness) is a payment or reward often offered by a group as an incentive for the accomplishment of a task by someone usually not associated with the group.
A bounty is a reputation reward you can put on a question to get it more attention for exactly one week.
- Stack Exchange (Stack Overflow)
Original motivation for bounties on Stack Exchange:
Speaking for myself, you don't have to reward me for ASKING questions on Stackoverflow: Getting an answer is enough of a reward in itself. Just do whatever you can to keep the answer-ers motivated.
- Corey Trager, quoted by Jeff Atwood, Reputation Bounty for Unanswered Questions
In each case, a bounty is some kind of reward offered for something. For Stack Exchange, that thing is an answer, and what's posted is a quantity of reputation.
Historically, bounties were often posted by governments with the intention of reducing criminal elements, and paid with real currency on a success basis. As there was no intermediary holding the money in escrow (the money remained with the government until the bounty was paid), there was no requirement for the concept of a refund for unclaimed bounties.
Today, the term bounty is also used in a sense similar to payments for service. The context has broadened, and so have the particulars surrounding them.
We now have bounty 'marketplaces', where one can post 'bounties' on causes and so on, without the stigma of criminality attached. For example, one can set a 'bounty' to raise funds for charity. They money involved in the bounties may be very little compared to government-funded bounties of old. Some marketplaces charge minimal transaction fees to cover their own banking fees, while others charge commercial rates.
With Stack Exchange, the context changes further. Bounties are paid in rep points, not 'real' money, and rep points have no financial value to Stack Exchange's organisation. As the first Stack Exchange quote above indicates, bounties are intended to be used as a form of advertisement. However, simply calling the 'bounties' advertisements would be inaccurate since advertisements are inherently transactions between the advertiser and the advertising entity - respondents to the advertisements have a separate relationship with the advertiser.
And this brings us to the heart of the matter.
Per the second Stack Exchange quote above, bounties on Stack Exchange were initially set up to motivate answerers. Regardless of the changes from the perspective of the people who post and administer bounties, to the recipient, it's a bonus they get for satisfying the terms of the bounty. To this end, the term bounty is used in a manner consistent with its historical usage, where it was also fundamentally a reward posted to elicit certain services from those who would ultimately receive the bounties.