I guess I'm not seeing where the confusion comes in. If you own a share of something, you earn a dividend.
You're "entitled" to it only in the sense that the holder of the investment is obligated by contract to pay it to you.
Alaska's Constitution states explicitly that all of the resources of the State of Alaska are owned by its citizens, not by the state government. Citizenship in the state has been determined by the state Supreme Court to be equivalent to being a legal resident, which requires 1 calendar year of residency, having a driver's license, etc.
If you meet that standard of becoming a citizen, which by definition means you have made a real investment in the future of the state by living and working here, then you have earned your share of the resources of the state.
With the influx of money from lease sales and oil production, the people of the state determined that one way of achieving that Constitutional imperative, in addition to removing state income and sales taxes, was to invest the money to run state government and pay dividends from part of the investment earnings.
So, you apply and you get the benefit you've earned by being a citizen of a state with ownership enshrined in its Constitution. That share is paid out in two ways. You don't pay taxes to the state AND you get a Dividend.
Additionally, the Dividend program requires that the residency not be just on paper. If you are out of the state for greater than, I believe six months, in any application year you lose the right to the dividend as you are no longer a "real" contributing member of the community up here (there are medical and military/public service exemptions to a point).
As Maned says, how is getting a dividend on your investment earnings of resources you own an "entitlement". You "buy the stock" by choosing to commit your life to living in Alaska, you "sell" the stock when you do any number of things to remve yourself from legal residency or the terms of the contract.