I have a gut feeling that it is, but I realise I’m in a vanishingly small minority (perhaps of one?). I have long held the opinion that percentage is merely a unit of measurement - the question in the title is like asking a length question and saying “What feet?” or “What metres?”.
The correct question should therefore be “What proportion?” and the answer may be expressed in whichever units the respondent wishes - in terms of percentage, or a fraction, or a ratio.
But I wonder if I’m being even more pedantic than I normally am.
Is it really like saying “What feet?” rather than “What distance?”, or is a) the grammar correct, and b) the questioner is merely specifying the units in which they want the answer expressed - the equivalent of asking “How many feet?”
But percentage is not a unit of measurement, no more than proportion is. There has to be a percentage of something, and that something would entail units of measurement.
I suppose it might be idiomatic in that it has just become a part of the English language. “What is the percentage?” would certainly not be wrong, but it sounds wordy.
What length is the coffee table? What depth is the swimming pool? What height is the bookshelf?
Those sentences seem to work.
What feet is the carpet? What pounds is the television? What teaspoons of salt should I add to the soup?
Those sentences do not work.
So perhaps the rule of thumb is that specific units of measure don’t work in that usage, but as is often the case with English, there are exceptions. This is why people learning English as a second language think we are all insane.
I don’t see anything wrong with the question. If I ask a question and would prefer the answer in a certain format, or units, it makes sense to ask it that way. If I am pouring you some coffee I might ask “how many sugar cubes do you want?” If I have a bowl of sugar cubes in front of me, I don’t want an answer in teaspoons, ounces or grams of sugar, I want it in cubes. I also thing when you ask for a percentage, the question conveys the idea of a large random sampling of something.
But the analogy for “feet” is not “percentage” but “percent”. Percentage <=> footage. IOW, “percentage” is not the measurement itself (“percent” is), but rather an expression of the measurement taken: “What is the percentage?” “10 percent.” “What is the square footage?” “20 square feet.” “What is the mileage?” “100 miles.”
So, yeah, “what percentage?” should probably be expressed as “what is the percentage”, just like we say “what is the [square] footage?” or “what is the mileage?” But I think it’s a generally understood shorthand.
OTOH, I think “What percent?” is perfectly correct, just as “how many feet?” or “how many miles?” is correct.