The mixture thickens up slightly when it cools, and quite a bit more if stored in the fridge. So if you want to use it as a vegan caramel sauce for an ice cream, let it warm up a bit at room temperature for a few hours. If you want to use it like a spread, then it's fine straight out of the fridge.
If you want this to be a very thick mixture, like if using inside of a cake layer or if you just prefer a thicker (non-runny) spread, you may want to thicken it even further by adding 1-2 tbsp arrowroot and 1/2 - 1 tbsp coconut flour in the last hour of cooking it. To add the thickener you'll need to make a slurry first so as to avoid arrowroot and flour clumps in the dulce de leche. The way to add it in is to first mix the arrowroot and coconut flour in a separate dish with a little bit of water (or more non-dairy milk). Whisk it well together until all the lumps have dissolved and then add this paste into your dulce de leche mixture and mix it thoroughly. The dulce de leche sauce will begin to thicken within a few minutes of whisking it on the low heat. Omit coconut flour if you want the mixture to look more gelatinized, and just use arrowroot alone.