Why is it easier for the ocean to push someone over by exerting force on their front side than by exerting force on their left or right side? If I put two clones of normal weight on a beach in the ocean, with one standing perpendicular to the waves, and the other standing parallel to them, the one standing perpendicular to the waves will be a lot harder for the waves to push over than the one standing parallel to the waves. Why is that?
Is it because there's more "stuff" from shoulder to shoulder than from bellybutton to backside?
 A: The main reason for this difference in stability is the orientation of the feet of the clones.  If you were to do this experiment on dry land by simply putting sandbags on someone's feet (the water makes it hard to move your legs fast enough to catch yourself, and so do the sandbags) and pushing on them from different angles, you would find that they fall over much easier forward and backward than side to side.  This is because gravity exerts more restoring torque about a foot spaced out to the side than about the ankle hinge, which is directly below the clone's center of gravity.
If the clones are standing within the point where the waves break, then there is also the effect that you hypothesized, that the flowing water of the wave exerts more force on the larger area of your front/back than on the smaller area of your sides.  This effect is relatively small, however.  Also, if the clones are standing in the swell outside where the waves break, this effect is hardly present at all, since most of the force toppling the clones in the swell comes from variations in the depth and pressure of the water rather than its lateral flow.  These forces are unaffected by the orientation of the clones.
A: It's the same reason a knife cuts through an object so easily. Use a blunt object and you'll have a harder time.
When a person is standing facing the waves or away from them, the force exerted on them by the wave covers a greater area all across them and therefore produces a greater net force on the object that would be their body. 
Standing on your side, your body is still receiving the same force but on a far smaller area and much of the water will slide right past you which leads to a far smaller net force on your whole body.
A: Generally, an object will tip over when its center of gravity moves outside of the ground-level contact area supporting it. 
Project a standing person's center of gravity to the ground, and you'll see it hits between their feet. Ordinarily, the CG has to move further side-to-side to the outside of one foot's support area than front-to-back past the heel or toe.
If a person brings their feet together, the support area will usually be a little narrower than it is long, and it'll be slightly easier to topple them to the side than front-back. 
If a person is wearing skis, the support area is much longer front-to-back than side-to-side, and it's much easier to topple them to the side.
A: Your answer "there's more "stuff" from shoulder to shoulder than from bellybutton to backside" is right. 
A mathematical rephrasing of what you said: the waves in the ocean exert a certain amount of force on each unit area that comes their way*.
Let's say you get 1 unit force for each cm^2 facing the waves. you will fill a much greater force if you stand perpendicular to the waves. 
A mathematical term that describes this nicely is flux, the amount of flow perpendicular to an area.
*this is because when water with velocity (or more physically momentum) collide with another body they give some of their velocity to that body 
