This is just one of those things that you will find in all languages – a phrase that developed a long time ago and is now set in stone. In addition to the previous answers ‘brake’ is not only a verb (= to brake) and “putting the brakes on” a colloquial form with he same meaning but in English the singular is also used as a generic technical term, for example if someone explains the function of it as a mechanical item (as in: “The brake is a device to slow down a moving vehicle.”). But otherwise the plural is used, probably because a car, like in your example, does not have just one brake but four, i.e. one for each wheel. That has also moved into general language. Here ‘putting the brakes on’ means simply to interrupt something or slow it down. You might hear, for example, “He worked all hours but after the heart attack he put the brakes on.” meaning he worked fewer hours and ‘slowed down’ his life style.

This is confusing because the word "brake" is a verb ("to brake" - "you need to brake earlier in the rain") but "the brakes" are a noun, they are a part on a car / train / bike / etc. Therefore either "you brake" or "you apply the breaks" Compare with "horn". You can "honk the horn", or "hit the horn" but you cannot "horn the stupid driver". "The brakes" are a thing. "A brake" is not a thing.

When you press braking pedal, 4 brakes will stop the car, because brakes install on tires.

I checked the Advanced Learner’s English-Chinese dictionary. It shows ‘brake’ can both be a noun and a verb. It also lists examples like air brake and disc brake. Besides, I checked COCA(Corpus of Contemporary American English). The matching strings shows brakes and brake are both acceptable.

It is a bit easier if you think of "brakes" as a noun, but "brake" as a verb. He put on his brakes to avoid hitting a dog. (noun) He had to brake suddenly to avoid hitting a dog. (verb)