There's no difference. You can use either one. 'Have' is a bit more common though.

In normal circumstances, I can detect no difference. Their meanings do differ in one situation though. Suppose you are my guest. It is time for breakfast. I am about to make breakfast for you. It would be polite for me to ask you "what do you eat for breakfast?". Knowing the answer (which would explain your dietary habits) would help me decide what to cook for you. In order to answer my question, you could then ask me "what do you have for breakfast?". The answer would tell you what food I have on hand. Knowing the answer would tell you what your options are.

Both options are commonly used and acceptable for asking someone about their breakfast. However, there can be a slight difference in emphasis. "What do you have for breakfast?" focuses on the items or food that someone typically includes in their breakfast routine. It implies a broader perspective, encompassing not only what they eat but also what they might drink or have as part of their morning meal. "What do you eat for breakfast?" specifically highlights the act of eating during breakfast time. It places more emphasis on the action itself and suggests a narrower focus on the food items consumed. Ultimately, the choice between the two options depends on your preference or the specific context of the conversation.

"What do you eat for breakfast?" refers to what you typically eat every day. "What do you have for breakfast?" would refer to what is on your plate at that very moment. Hope that helps!

Both questions are correct use whichever is most comfortable for you.