There is a science to "stamina." Endurance athletes have been studied at the cellular level for nearly 40 years, going back to Bob Costill's studies at Ball State. By that, I mean that they have had their muscle cells sampled at rest and during exercise, as well as blood volume, gas exchange, and everything else you can imagine. These results have been matched with performance over many years. If you want to know about "stamina," which I interpret as "endurance," then there are multiple sources. In a nutshell:
1. Genetic muscle composition is important. We all have a fast-twitch/slow-twitch mixture of muscle cells. Those with higher slow-twitch compositions tend to have greater endurance, although fast-twitch muscles can be trained to act like slow-twitch muscle cells.
2. Genetics play a role in energy efficiency. Some people have more efficient mitochondria, and therefore process more oxygen to make more ATP; hence, some people have better endurance by genetics. However, endurance training greatly increases the size and number of mitochondria in a cell, and over time the training will overwhelm the natural genetic advantage some people have. A trained athlete will always outperform an untrained person who happens to have good genes. In the first year of long-distance running, an untrained person will increase his or her VO2 max (the amount of oxygen processed during exercise) by 100%.
3. There is definitely a "mental component." Various models are hypothesized, but one of the recent ones is the "mental governor." This model suggests that every person has the ability to essentially exert himself at full speed until his heart explodes like a horse, but our brains tell us to stop, like the governor on a car's speed. Elite athletes train themselves to push beyond what their brain "allows," with the idea of "resetting" the brain's governor.
The Bottom Line: Any motivated soccer player can develop outstanding endurance with mental and physical training, far outstripping those with "natural" endurance, but who do not have a long-term training plan.