There are lots of potential thickeners, but you often need to select the one that works best with your given need (temperature, if it has dairy, resulting mouthfeel, etc).
In your case, you're already using rice, so you may want to stick with a starch -- corn starch, potato starch, tapioca, etc.
For these, you add a bit to cold liquid, mix it well, add it to the soup and heat it up.
As it approaches boiling, it'll thicken.
You can use flour (wheat, rice, yam, etc.), but in the case of wheat flour and possibly the others, you may have to deal with a raw taste.
You fix this by cooking the flour with butter or oil into a roux, and then adding that, and heating it up to a near boil.
You can also use pureed vegetables.
Cook them in the soup, then put it through a blender so the vegetables add body to the soup.
cooking.stackexchange.com/users/67/Joe
If you are looking for the basic and common ingredient to thicken the soup then I would suggest to go with Corn flour.
All you need to do is mix corn flour in cold water.
Add it into soup and stir it well.
And you are done.
cooking.stackexchange.com/users/22295/The Hungry Dictator
@Joe has provided an excellent all-purpose answer, but there's one possibility I don't see mentioned: egg.
A beaten egg stirred into your rice mixture will bind and help thicken things up.
It will also make your dish a bit richer.
cooking.stackexchange.com/users/45428/senschen
This is just an opinion.
I am not in too keen on using potentially gelatinous materials like corn starch or gums in this specific situation with coconut milk and rice.
Unless I had misunderstood your intention, thickening the excess liquid will only completely alter the texture.
It is a bit like you having put too much milk into your breakfast cereal, thickening the milk is not going to work "normally", you need more cereal.
So, more rice is the real solution.
Even if you wanted to improvise and make the liquid thicker, I would want to use rice flour or blend some cooked or uncooked rice with excess coconut milk taken from your pot, and then use it as your "invisible" thickener and cook the whole pot for a short while until you get the desired viscosity.
cooking.stackexchange.com/users/49834/user110084
Just drain it - treat the rice like pasta.
When it is done cooking, use a fine mesh strainer to drain the excess liquid.
I can't find an online source, but I remember (hopefully accurately) reading a James Beard cookbook where he suggested this as a general method for cooking rice.
cooking.stackexchange.com/users/7893/Gregor Thomas
Yesterday, I had some cooked rice and poured in coconut milk. The can turned out to be bigger than expected so the contents got a bit too "loose".  One way was to cook some more rice, of course. However, I wonder, is there another commonly usable ingredient to thicken up things?  I'm thinking - if it's too thick, I'd pour in some water or milk. Possibly oil too. But what if I want to accomplish the opposite?
Common ingredient to thicken the soup
cooking.stackexchange.com
<soup><thickening>
cooking.stackexchange.com/questions/81615
cooking.stackexchange.com/users/57839/Konrad Viltersten
You could just drain the rice like pasta, but when thickening you should consider compatibility with the purpose of the dish. A starch like corn starch, potato starch or tapioca, or wheat, rice, yam or corn flour could work, or you could add pureed vegetables or a beaten egg to halp thicken things up. There are two ways of dealing with rice that has too much liquid. The first is to thicken the liquid with corn starch, flour, pureed vegetable of beaten egg. The second is to drain the excess liquid from the rice.
There are a number of thickeners available. The one to use depends on the food you are cooking. The options include corn starch, flours, pureed vegetables and egg. An alternative way to remove excess liquid is to drain the rice.