I often like to equate shares in the game to victory points: you need to buy 10 * the number of opponents to win. The performance of yourself and other players determines how expensive those points are to get, and in a pinch you can sell points back at their current value (hopefully at a profit compared to when you bought them).
They're not like real shares, because when you buy someone's stock they aren't getting money for it (that I'm aware of). The implication is that they've already been issued and you're buying them off the interplanetary market (I assume that your starting cash and assets is the result of that initial offering). They double in price when you are buying shares that a third party owns, for several reasons:
1) If players could choose to sell their stocks in other players at all times, then you might reach a deadlocked position where no one's interested, regardless of offered price (in reality there would be a breaking point, but this is a game where a winner must be declared).
2) By increasing the price of pre-owned shares, you make it dangerous for someone to buy someone else out of ownership, rather than just being a kingmaker of sorts - you're giving the person you AREN'T removing from the game a chance to get the advantage with the extra money you're giving them.
Of course, this doesn't stop two players ( A and B ) from colluding and having Player A simply sell their shares in Player C so that Player B can buy them at normal price. The only difference between this and the normal buyout process is that it can be done at any time and not just when a player is being eliminated, and you aren't giving extra money to the selling player.