The Mario series of games, and their basic story principles, have a very sexual theme. The driving force behind the plot is sex based, as is all of the higher level innuendo hidden in the game.
The game operates on the Bowser steals princess, plumber wins princess back concept. To further explain this, one must identify just what makes the character of Bowser the antagonist monster that he is. One might think this fairly obvious, as he is a massively overgrown, green, reptilian tyrant, with sharp teeth and spines, but upon closer inspection, these graphical qualities all connect to the cultural idea of an antogonist. His being overgrown can be seen as gluttony; his being armed with teeth can symbolize violence. Reptilian forms denote a lack of emotion, apathy, and also a departure from what is considered beautiful and sexual. To put that another way, Bowser is designed to look like the antithesis of Mr. Right.
This is where Mario and Princess Toadstool come in. They perpetuate the cultural theme of the ideal couple. As a plumber, Mario is very masculine, hardworking, and strong. As a princess, Toadstool is very feminine, receptive, and beautiful. Together, they form an ideal couple.
So, one must wonder, what is Bowser's motivation for kidnapping the princess? It isn't money; he has several castles. It isn't power; he is already a tyrant. The only thing Bowser doesn't have is applicability to the ideal couple. However, if he is married to a princess, then he will have that. So, his motivation in stealing the princess is to bring himself up to Mario's status as the ideal male in the ideal sexual relationship. And thus, the Mario story is intrinsically based on a sexual conflict.
This opens up an entire avenue of innuendo. Why is Mario a plumber? Because that means he is good at "laying the pipe," or even, "working with her piping." As I quote from amib's definition of Mario, he, "gains powers by eating or touching various objects, and attacks enemies, traditionally, by leaping into the air and landing on them." This innuendo is so deep in the game that to defeat Bowser, Mario must cut off his long bridge with an axe.
A quick chronology of the Mario story: Bowser steals princess, and spends the entire game alone with her deep inside a castle, then Mario castrates him, by destroying the bridge, and wins back the princess. Thankfully, players of the game merely need to press the reset button, and everyone is a virgin again.