Homemade coconut milk separates into different layers after a day in the fridge
All coconut milk separates out. When you buy it tinned, there is often a plug of coconut cream above a pocket of coconut milk. But it certainly should recombine. It may well have been too cold when you tried to stir it together - let it come to room temperature at least.  If you warmed it up on the stove you'd find the two would melt together without any stirring necessary.