The most obvious examples that come to mind are from the Bible, where pharaohs are never named:
“I cannot do it,” Joseph replied to Pharaoh, “but God will give Pharaoh the answer he desires.”
Maybe the actual names of the pharaohs in the Bible were lost or omitted because they would sound too foreign to their intended audience. But I've definitely seen it used that way in other contexts (EDIT: see examples below), and you'd never see the word "king" used this way even if their name was unknown.
Apparently "pharaoh" originated from a word meaning "great house" but that doesn't seem to really shed any light on things. Neither did this Wikipedia article on Egyptian royal titles.
The Bible even includes a kingly title after the name Pharaoh in a number of instances, e.g.:
“Go, tell Pharaoh king of Egypt to let the Israelites go out of his country.”
I'd also be curious to know if there are any other cultures with similarly-functioning titles.
EDIT: Found some non-Biblical examples from a Futurama episode. Transcript here - it's used frequently. Would this just be a convention they picked up from the Bible?
We can safely assume that the use of Pharaoh in this way derives originally from Exodus, whether from that story specifically or from the cultural context in which it was written. Pharaoh is a direct transcription of Hebrew פַרְעֹה, which (as you've said) is a borrowing from Egyptian 𓉐𓉻, "great house".
The use of the phrase "great house" to mean "king" is a straightforward case of metonymy (using a thing to refer to a related thing). It's analogous to the way we use the phrase "The White House" to refer to the the president of the US. This happened in Egyptian as well. It began long before the Exodus story was ever composed and continued for the rest of Egyptian history. The Coptic word for "king" is ⲣⲣⲟ (rro), and "the king" is ⲡⲣⲣⲟ (prro) (the p in prro was later rebracketed as the definite article). So there's nothing strange about the word itself. It was the actual Egyptian word at the time.
I do know of at least one case of the Egyptian word pharaoh being used as a name explicitly. On the Temple of Dendur in the Met, there are cartouches with only the word Pharaoh written in them, which makes it a proper name. This is much later than the events or story of Exodus, however.
A great deal has been written on the historical kernel in the Exodus story. (This is my favorite hypothesis so far, but you can easily find many others.) In short, the story cannot possibly be literally true in every detail. It's a "charter myth"—a story that explains a people's social system to themselves. As such, it does not need to be literally true in every aspect, but it does need to maintain verisimilitude.
One of the ways the story strives to ring true to its audience is by using overtly Egyptian words and phrases borrowed into Hebrew. Pharaoh is one of these. The story is in Hebrew, but it includes the Egyptian word as the name of the pharaoh. The story doesn't actually care who the specific pharaoh was, because the story itself is an oral legend rather than a precise history, but it does want you to feel that these events happened in Egypt. It's a bit like writing an English story with a French king named Roy (and pronouncing it "rwa"). It gives the story a certain flavor.
