Does the shape of the bushing matter?

Yes it does! The bushing is a very imported part of the truck and impact the way of steering and feel a lot.
These are the most common bushing shapes out there:

Barrel and Tall Barrel - The perfect fit for most of the bushing seats. Barrels will excel in downhill and freeride, where you want your trucks to turn and return to the middle position with some rebound. Softer barrel combos can be nice for cruising, however, if you want to make your set-up more turny you can try running barrel board side and cone roadside. For higher speeds, we recommend a barrel+barrel combo. Some trucks are designed for tall barrels, taller bushing creates more space for a truck to move. Therefore you can get more turn from your trucks. However not all trucks fit tall barrels, you should always check if your truck can run tall barrels as if you put them where they are not supposed to, you can mess up with truck geometry and whole feel.

Eliminator - Designed with high speeds in mind, eliminator is supposed to eliminate speed wobbles. It also restricts the truck from turning too far, so sometimes they can be used as wheel bite stoppers. Usually run board side, because of their big size, you can run softer duro than you normally would.

Street Bushings - This could be a category on itself, but basically TKP trucks are designed with shorter bushings in mind. Take a look at bones bushings, with their special, no washer technology. You can change street bushings if you are looking for tighter or looser trucks, and you can find similar shapes in longboard bushings, so again double cone will be more turny than double barrel.

Cone and Tall Cone - Less restrictive, with less material used than the barrels, the cone will allow you to carve and change directions quickly. Cones are less stable at higher speeds, but perfect for lower speed cruising and carving. Tall cone is a little bit more restrictive but still delivers that playful feel. For the smaller kids we advice a double cone bushing combo. It has almost no restriction at all and allows kids to turn with ease.