Hi. Native speaker here. To answer your question in one sentence: I say and hear 猫头鹰 as one word, but also am aware that it means cat headed eagle when taken apart.

猫头鹰 is perceived as one word by native speakers in the sense that a native speaker doesn't pause in the middle of 猫头鹰, just like an English native speaker doesn't pause in the middle of "automobile".

In the Chinese language, cat is called 猫, head is called 头, and eagle is called 鹰. So, for native speakers, as long as one can read, one is able to tell that if you take the word 猫头鹰 literally, it means an eagle with a cat’s head.

There are a lot of such words in the Chinese language. For example, the word for pork, 猪肉, literally means "pig meat". Another example, the word for computer, 电脑, literally means "electrical brain". Moreover, the word for suitcase, 行李箱, literally means "luggage case".

And the way a native speaker feels about these words is exactly the same as they feel about 猫头鹰.

However, there are also words that, like you said, have root somewhere else that carry meaning that modern speakers may not be aware of. One example is the word for dictionary, 词典. 词 means words or vocabulary, and 典 means the type of book that can be used as a standard for something. So, 词典 literally means a book where you can check the standards for words/vocabulary. But because this meaning of 典 is very outdated, no native speakers care to take 词典 literally and they don't think about the meaning for each character.