Is "drunken boxing" a real martial art or only Hollywood fiction?

It has its roots in a genuine Chinese martial arts concept, in which the fluid movements of a drunken person are adapted into this real skill.

"It is said that Drunken Boxing was derived from the fighting skills used by Wu Song, one of the characters in the novel Outlaws of the Marsh, when he beat a hooligan surnamed Jiang after getting drunk, as well as the attacking skills used by Lu Zhishen, also a character in the novel, when he caused an uproar in the mountain as he was drunk. The movements of Drunken Boxing are guided by the principle of "drunken in appearance but not in spirit".
https://en.wikipedia.org/wiki/Drunken_boxing

"The postures are created by momentum and weight of the body, and imitation is generally through staggering and certain type of fluidity in the movements. It is considered to be among the more difficult wushu styles to learn due to the need for powerful joints and fingers. Zui Quan is sometimes called Zuijiuquan, (literally "Drunken Alcohol Fist").
https://en.wikipedia.org/wiki/Zui_Quan

In Zui Quan or the drunkard boxing, boxers falter, waddle, fall and sway just like drunkards. Zui Quan can be used for both fighting and maintaining health. However, the drunkard boxers go out of their way to stress the combative side of their style. They blend a series of movements, actions and skills of the martial arts and try to confuse their opponents with special skills which often lead them to surprise triumphs.

The main feature of the drunkard boxing is to hide combative hits in drunkard-like, unsteady movements and actions so as to confuse the opponent. The secret of this style of boxing is maintaining a clear mind while giving a drunken appearance.

Drunken Boxing' is an enigma in Chinese Martial Arts. There is no stand alone "Style" of Drunken Boxing; this is only in the movies. Only a few systems have a true 'Drunken Boxing' set and that set is usually the System's most guarded secret. Choy Li Fut is one such system.

https://www.liveabout.com/style-guide-choy-li-fut-kung-fu-2307676

Then a true Drunken Boxing Master would not be intoxicated when he engaged in self defense.

Well, let's put it this way...he wouldn't need to be.