It would sound ambiguous if you didn't use 'tane'. Kaç is actually a question word that can usually be translated as 'What' but it's specifically used for numbers. Some examples:
Numaran kaç? - What is your number?
Yaşın kaç? - What is your age?
Saat kaç? - What time is it? (lit. What is the time?)
Elmaların sayısı kaç? - What is the number of the apples?
Bunun ücreti kaç? - What is the price of this?
Beside this, you can use 'kaç' as an interrogative adjective. (Interrogative adjectives are questions words that are used like an adjective i.e. Which car?) So in that case it would clearly mean 'how many' because that's the only meaning you can get from it as an adjective and you don't need to say tane. For example
Kaç (tane) elma var? - How many apples are there?
However, if you only say 'Kaç?' it would be ambigious because it can either mean 'How many?' or 'What is the number of it?'
Elmalar kaç? - What is (the number/price of) the apples? or How many are the apples?
So, it would be enough to use kaç without tane only when you clearly use it as an adjective (i.e. Kaç elma var?). Otherwise, people stick with using tane to sound more clear.
Not to confuse anybody, but it's worth mentioning: "kac" is also a verb meaning to run away. The two are in no way related, but it may come up. Kac! Kac? All explanations for the question form of kac are good, but missing one thing. You would almost never hear the "kac" in its question form used alone. It will always be followed by the subject in question: kac para, kac ev, kac kisi, kac tane.