Is the water you drink the same water you urinate a short time later?

No the water you drink has to go through a multitude of body filters before it is eliminated from the body.

The human body contains lots of water. When you drink water it gets added in to the water you already have. When the kidneys remove water from the blood, that water will include some water that you recently drank, but also some water that you drank at some earlier time. In addition, not all the water in the human body is obtained by drinking. The metabolic process by which energy is obtained from sugar (or nutrients which become sugar), also produces a certain amount of water as a by-product. And of course, many solid foods have water content as well. So water in the human body comes from various sources.