There's no one path in terms of education and professional experience that leads to management consulting. For example, I studied International Relations, French, and International Economics in my undergraduate and graduate studies and began a career in international development and microfinance. Later in my career, I went back to business school and pivoted into strategy consulting.
I agree that speaking to different companies will give you a sense of what they are looking for and that analytical, critical thinking, and problem solving skills are important. These skills can be cultivated regardless of your background and can be leveraged to help solve clients' problems in industries that can be quite different from where you came from.
For example, having experience in international organizations and nonprofits, I found myself as a consultant in the financial services and pharmaceutical space.