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a.) This animal is half goat and half sheep.
b.) This animal looks half like a goat and half like a sheep.
In sentence a, half does not need to be followed by an indefinite article, does it? But why? In sentence b, "goat' and "sheep" each have an indefinite article. Why the difference? How do you analyze a?
This animal is half goat and half sheep
When you use "is", this means that the animal in question bears some permanent traits that will either be considered a sheep or a goat based on different aspects. Maybe it's a goat-sheep offspring: it has both the genes from a goat and a sheep; or maybe one side of it is a goat, the other side is a sheep, kind of like what you'd get if you chop both a sheep and a goat in half and join each half together; etc.
Sidenote: when phrased this way, this sentence is really vague and it's very hard to pinpoint the exact meaning the speaker is getting at.
This animal is half a goat and half a sheep
When phrased this way, personally, I'd consider it to lean more to the latter meaning (one side of it is a goat, the other side is a sheep, kind of like what you'd get if you chop both a sheep and a goat in half and join each half together); but it's still quite similar to the above sentence.
This animal looks half like (a) goat and half like (a) sheep.
In this sentence, I feel like the articles can be ommitted without significant change in meaning. To me, "a" just feels like a meaningless filler without conveying much meaning in numbers.
Unfortunately, I have no other sources other than my ear.
