A cup of tea says about the beverage than the cup. A tea cup is a cup that's used to serve tea. A cup of tea is a tea cup with tea in it. The same goes about other similar things like a glass of beer and a beer glass.

The word "tea" is spelled with an "a." "Teacup" is one word, with no space in it. Words and phrases like "teacup" have customary meanings. Logically there could be more than one meaning. Logic does not tell you the meaning. You just have to learn the meaning. A "teacup" is a small cup made of porcelain or china. It is a good size for drinking tea. A "cup of tea" is a teacup that has tea in it. A "paper cup" is a cup that is made out of paper. A "plastic cup" is a cup that is made out of plastic. A "foam cup" is a cup that is made out of plastic foam. A "coffee cup" is a cup made out of porcelain or china. It is a good size for drinking coffee. In a restaurant they serve you coffee in a coffee cup. After you drink the coffee, they wash the cup and use it again. In a coffee shop they serve you the coffee in a paper cup. The cup is full of coffee. The cup is made of paper. Paper is cheap. They give you the paper cup. You take the paper cup and coffee with you. After you drink the coffee, you throw away the paper cup.