A king also has a higher level of kedusha because he has certain mitzvos that apply specifically to him. For example, he is restricted as to the number of horses he may own and the number of wives he may marry. There are rules that govern his appearance which must be distinctive. (Note: the use of special attire denoting royalty is a an important theme in Megilas Esther, as Haman in his desire to attain the throne specified it as the first gift to be given to one that king seeks to honor).
Indeed, the Tosefta states specifically that a king has an added dimension of kedusha. This added kedusha comes from the anointing process and the appointment to his position by the Sanhedrin. The leader of the Sanhedrin also has a special kedusha. The Rambam (Hilchos Sanhedrin 26:1) includes the nasi, head of the Sanhedrin, and the king among the list of people that one may not curse. Like the king, his appointment to his role grants him an added level of kedusha.
These higher degrees of kedusha, (for a king, kohen gadol and nasi), are all rungs in the ladder of kedushas Yisrael. They all require minuy, appointment from an external source, as a pre-requisite to attaining the role. These individuals require a uniform to remind them that they have been appointed by the people to represent them, and the additional mitzvos that apply to each of them grants them their elevated kedusha status.
(Note: see the Rambam, Hilchos Mlachim 9:1, who distinguishes between the generations from Adam till Moshe based on the additional mitzvos each successive individual was commanded to perform. While Chazal tell us that the patriarchs observed all of the Torah prior to mattan Torah, perhaps they did so from the perspective of one who voluntarily performs a mitzvah without being commanded to do so. Chazal teach that there is a higher level of performance, perhaps kedusha, ascribed to one who performs the mitzvah in response to the command to do so. Perhaps the patriarchs acted as metzuveh v’oseh for those mitzvos that they were directly commanded to observe, the observance of which increased their level of kedusha as compared to their predecessors. They were in the category of eino metzuveh v’oseh for the rest of the commandments that they observed voluntarily.)
Moshe Rabbeinu did not require his appointment to be sanctioned by bnei Yisrael. His inner personality anointed him and sanctioned his roles as kohen gadol, king, judge and teacher. The statement that Moshe wore chaluk lavan indicates that Moshe was above appointment by the people. After all, how could a uniform describe his status as the greatest of all men, the individual selected by Hashem to be the direct recipient of the Torah and the one entrusted with the task of transmitting it to bnei Yisrael?
Moshe, who was constantly in a state of lifnei Hashem, always prepared to encounter Hashem, so to speak, did not require external symbols to sanctify him. Moshe did not wear bigdei chol, demonstrating that his kedusha was of a different type and level relative to the rest of bnei Yisrael. He also had no need for bigdei kehuna, since he was beyond the need for a minuy and was able to function as a kohen gadol without an external appointment or sanctification process. His kedusha was also distinct from that of the kohanim, including the kohen gadol. He wore a chaluk lavan, something that was unique, as he was.