At the time of US independence, the 13 colonies had 2+ million inhabitants, whereas the mother country, Britain, had only 8+ million.
To be slightly more precise, the figures are probably 2,750,000 for the newly formed USA and about 7,5 million for Great Britain, excluding Ireland (around 4 million).
How is it possible a colony could grow organically to 25% of home population in such a short time?
That's mainly a matter of math. Essentially, it's a relatively small disparity between the birth rate and the mortality rate in the colonies which, over a period of about 150 years, resulted in marked growth. Whereas those figures in Britain remained much closer to one another and hence resulted in a less spectacular population increase.
The growth rate of the English colonies was remarkable. Especially in New England, which saw spectacular increases in population without the need for immigration. The annual population increase in New England exceeded the death rate by well over 3%. This might seem a small number, but figures like these are very significant when they're sustained over a long period of time because growth then effectively becomes exponential.
For example:
A population of 1,000 people (women) with a constant annual growth of 3% will result in a population figure of about 19,218 a century later; which is an increase of well over 1800% compared to the initial 1000 people. Based on commonly cited estimates and without taking additional immigration into account, the 1780 US population had increased by less than that figure (increasing by about 1735%) over the preceding century. When accounting for immigration, an annual growth rate of 2.8% will easily get you from 151,500 colonists in 1680 to about 2,780,400 Americans in 1780.
To again stress the huge differences in outcome regarding these deceptively small figures: had the net growth rate been 3,5% instead of 3% the 1680 population would have resulted in 4,725,450 Americans in 1780; i.e. almost double its most probable historical estimate.
By comparison: the historical population of Britain is estimated at 6.5 million in 1680 and 7.5 million in 1780. This is a significant 15% increase, but a far slower growth rate than that of the Thirteen Colonies. The average birth rate and death rate largely explain this. In the Thirteen Colonies the birth rate lay between 6.5 and 7.5 for much of the 17th century. Whereas the average total fertility rate in 17th century Britain was about 4.5 per woman. At the same time the average mortality rate in Britain (4.0-4.5 between 1640 and 1750) was much higher than that of the colonies, accounting for a slower level of growth prior to 1750.
In a large part, the discrepancy in death rates can be attributed to living conditions. 90% of Americans were farmers, living on countryside farms or small villages. Apart from a few port cities, urbanization within the American colonies was largely non-existent. In contrast, close to 35% of the British population lived in settlements with more than 2,500 inhabitants, with 15% of the total population living in cities of over 20,000 people. In general, for any period prior to the advent of modern sanitation and medicine, cities and large towns have a negative effect on the average health. It's hard to quantify this with solid examples, but one would be that of average height. A sample of British males (prisoners on their way to Australia) in 1780 showed the average height to be 5 feet 4 inches, whereas a sample of soldiers from the American Continental Amy in 1776 measured 5 feet 8 inches, showing they were about 10 cm taller on average.
Poverty was also less of a factor in the ever expanding United States, with more than enough "free" land for sons of farmers to settle on; essentially providing a basic way of supporting oneself for everyone willing to work the land.
American population growth was mainly, but not exclusively due to organic growth. Some 400,000 people immigrated (whether by their own choice, or not) to the Thirteen Colonies from Europe and Africa between 1700 and 1763. Even though immigrants, for most of the period prior to the American Revolution, never accounted for more than 10% of the population, they nevertheless contributed significantly to the ever expanding population base. Additionally, in the 17th century (when 85% of the immigrants came from Britain) these migrants slowed down population growth in Britain, though these effects had lessened by the 18th century, when less than 20% of the newly arriving immigrants were of British origins.
The period just following the initial colonization efforts in 1620-40 saw the start of the English Civil War, which (as a civil war) had a larger more negative impact on British population figures than a standard war; fought by abroad. Its overall impact on the population however, was still a relatively minor one.
No other colonial power faced the same power disparity.
Yes, but only when concerning the heritage of the settlers. Spain proper made up 42% of the total population of the Spanish Empire in 1800. By comparison, the Thirteen Colonies made up 'a mere' 27% of the combined population of the colonies and Britain in 1780.
It is true, that the ethnic make up of the Spanish Empire was far more diverse/mixed than that of the English colonies and that they also included large percentages of native populations. The main factors in this difference in population composition are natural ones. The Spanish, for example, colonized latitudes with harsher climates and relatively more tropical diseases, for example malaria. The English colonies in North America, by comparison, were very similar to Britain itself; allowing for a rapid colonization by farmers and requiring far less adaptation in terms of agricultural methods. The same process, on a far smaller scale, can be observed in sub-Saharan Africa, which has no significant (genetically) European population beyond the descendants of the Dutch in South Africa; which is the only locality with a Europe-like climate.