Student or Learner
As an isolated statement, why is it incorrect to say this?
I am going to meet my cousin brother.
Why is the usage of brother after cousin incorrect?
I want to indicate the gender on my cousin (brother and sister) in the same sentence, for that I use brother, so what is incorrect in this?
The phrase "cousin brother" makes no sense, as a cousin is the child of your aunt or uncle, and a brother is your sibling with the same parents, or at least one shared parent.
If you want to indicate sex, as bhai suggests use the person's name, or say my male/female cousin.
There are jokes, involving incest, about this matter. Here in Pennsylvania mean people will say of West Virginia (a mountain state to the south) that they have an unusual way of pronouncing "wife" -- \kuzin\.
In India "cousin brother" and "cousin sister" are common ways to describe one's cousins, I see nothing at all wrong with it. There are 150 million fluent English speakers in India, Indian English is a recognised dialect as is American English. Please don't mock it.
I can't imagine how you would think I was mocking you -- if that's what you meant.
What I said about West Virginia I meant as an example of how people can be cruel. It must happen in other parts of the world.
The straight answer to the serious question was already supplied by those who commented above.
About the English of India...I am far from critical, but rather an admirer.