Allegorically, spiritually kingly attire represents one with authority and who could be a better person than Kalki avatar himself, a man exercising absolute authority on mother earth.
A sword allegorically reflects power of Dharma (righteousness). To root out evil from the society in totality, the ruthlessness displayed by Kalki avatar is reflected in a sword. When it is time to deliver justice, none can come in way of Kalki avatar not even judiciary, Administration or policing.
The white horse displays power of absolute truth reflected in character of Kalki avatar. The truthfulness displayed by Kalki avatar would be phenomenal, without precedent in recent times. This truthfulness would be reflected in teachings of Bhagavad Gita and Upanishads imparted by Kalki avatar.
One becomes knower of all only when one was totally truthful from the beginning of life. Kalki avatar would be an individual who carries with him power of truth, power of Dharma and also full authority that could ever be exercised by a human being.
To save lives of millions Chanakya (the most able administrator in the entire history of mankind) got eliminated 100 kings in one night. Kalki avatar would be a Chanakya coupled with a self realized soul. Such monumental power would be exercised by one in coming times.