The description for the fog cloud spell says:

You create a 20-foot-radius sphere of fog centered on a point within range. The sphere spreads around corners, and its area is heavily obscured.

Is it possible for my wizard to cast a green fog cloud during a battle against goblins, and later in the same day cast a red fog cloud when fighting orcs? This would signal to a distant observer what the opponents are in the area.

If I know my opponent lost his family to a cloudkill spell, may I create a fog cloud that has yellow-green fog like the cloudkill spell? Or if my opponent fears a magical violet mist with gold sparks, may I cast a fog cloud with those effects?

My intent here is to understand the limits of the existing spell.

The Fog Cloud cannot be colored, RAW

Without circumstances dictating otherwise, spells in 5th edition D&D only do what they say they do, and no more. The Fog Cloud spell says it creates a 20' radius sphere of Fog, and that's all you can guarantee it'll do. If you want it to do more than that, you'll need a different spell.

You should be able to combine Fog Cloud, Light, and (optionally) Mage Hand to create the appropriate effect

Placing a colored light source inside the Fog Cloud should diffuse the light through the fog, creating the colored Fog Cloud of your choosing. Since of those three spells, only Fog Cloud requires concentration, it's possible for the same spellcaster to maintain all three effects simultaneously, and both Light and Mage Hand are cantrips, meaning you're not spending extra resources (other than your Action Economy) to produce this effect.

Simply cast Light (for the color of your choosing) on a rock, then use Mage Hand to lift the rock into the fog cloud (or simply place the rock inside the fog cloud if your character is physically able to do so). The range on Light extends at least 20 feet, and while it wouldn't disable the Heavily Obscured effect, it is plausible that a character could generate a colored fog cloud using this method.

Bear in mind that you may need a permissive DM to make this work, RAW, since the fog will surround the light, and while this should not obscure the light source itself, it's not clear that the physics of the light diffusing through the fog will result in the fog itself taking on the light's color. So it does require your DM to allow for real world physics to take precedence in this situation.

You could try to obtain a similar effect just using dancing lights. Since you have four of them you could move them all up in the air into a prearranged configuration to indicate the presence of a certain monster.