I did think of the milk example, but I’d argue that’s not quite an apt comparison as milk is sort of a generic term for similar things from different sources and often just any white liquid. If we had a different, special word for just cow’s milk for example, it’d be weird to call anything else by that name.

I’d argue the equivalent term to “milk” in this case would be syrup. There are many types of syrup, but we specifically use “honey” to mean “syrup made by bees”.