So most of the time if a character is a specific color, that signifies something about their personality or character arc. Not always, but most of the time.
So these are the typical connotations associated with different colors. Some of them anyway.

White - white can symbolize purity and kind intentions, but can also suggest aloofness and lack of empathy, or occasionally lack of personality, it often symbolizes truthfulness, new beginings, honor, and order. Can be associated woth the purity and cleansing of death.
All original unicorns were white.

Pink - a sign of sickness or old age in most species, as well as occurring most rarely in nature, pink therefore has a stigma of weakness, frivolity, and stupidty. This isn't often true, but exists nonetheless. Hot pink, however, can be seen as a rebellious color, but usually is considered arrogant.
The idea of pink unicorns was started as a mockery in many human cultures. To this day, no pink unicorns are known to have exsisted.

Red - unlike on earth, red is not often associated with anger, it is seen as a bold, lively color, symbolizing a rebellious nature and desire to live. Conversely, it is also seen as the color of death and barroness. Usually the lively and somewhat friendly aspects of this color outweigh this. The color most closely associated with life.
Dragons often refuse to wear or paint themselves this color as they fear it makes them look too approachable. 

Orange - A happy, lively color like red, orange has more juvenile connotations, and is rarely seen on powerful creatures. Orange can symbolize forgivness, as well as affection towards another person, often romantic. It is considered the color of spring.
In some cultures a formal symbol of forgiveness is to send an orange fruit to the offender.

Yellow - the main color of affection and simple happiness, yellow can occasionally be a negative color, but the purest yellows are always uplifting, yellow gifts alternatively suggest platonic or sometimes familial affection, as opposed to romantic. Yellow occasionally has opposing effects, as it is sometimes seen as lucky or a sign of death. 
Yellow flowers are often placed or grown around the home of a young child, and are said to bring luck for many humans. In some cultures, they are also planted around the homes of the deceased.

Green - green is in many ways the most diverse of the colors on this list, the types of greens vary these meanings immensly. Green can be seen as poison or pain, as well as a sign of confusion. It can symbolize youth and fresh life, but also confusion. The color most commonly associated with creation. Green can be calming or a very spirited color. Also associated with wisdom, peace, and life.
Though very few creatures are green, most planets have either rocks, soil, or something akin to plant life in this color.

Blue - blue is not a sad color, rather is considered to be aloof and wise, and can symbolize large gaps in space, the color of emptiness. Also a color of peace and prosperity, blue can symbolize a wealth of knowledge, material, or power. Blue is said to be the color of a journey.
Blue is a very powerful color and most of the Arcadian's armor is blue strainium.

Purple - A color representing time, power, and knowledge, a favorite of royalty and dragons. Ancients are typically associated with this color, and it is considered to be magnificent and strong.
Purple unicorns are surprisingly common.

Black - black is the color of sadness and doubt, but also of warmth and promises. Black is a powerful color, representing strength and mystery, and sometimes emotion. Black can be a color of anger, as well as intimacy. Black is considered a passionate color, and the type of black can represent an individual's interest in the proceedings. Black is seen as lifegiving and comforting.
Though most dragons are black today, they were originally colorless.

Brown - A color that symbolizes depth and reality. The darker the brown, the more hidden that depth is. It is also seen as warm and lifegiving. A fairly common color.
Brown is a symbol of fertility in many cultures.

Gold - gold can be a harsh, unfamiliar and unforgiving color. It can also symbolize warmth and joy, particularly for others. It is strongly associated with tradition.
Gold is not a particularly valuable metal on most planets.

Silver - silver is a symbol of lofty ambitions, disconnection, and elevation. It as seen as a color for the inherently lucky and well off. A high ranking color. It also symbolizes change and flexibility.
Unicorn officers are distinguished by a silver plate on their chests.

Bronze - A sign of stubbornness and strength, can be a sign of change or refusal to change. The color of revolutionaries.
Many species use bronze colored armor.

These are only the basic colors covered here, some are invisible or nonexsistant to humans so that's difficult. Also it is not so much totally different colors or that something doesn't have color for humans. Unicorns just see so much more.
Anyway I hope this was helpful. Enjoy.