Yes, it is possible to convert a consulting internship into a full-time position. Many consulting firms offer internships as a way to introduce students to the consulting industry and give them the opportunity to gain practical experience. Think of it like an extended period of getting to know each other.
These internships often provide the opportunity to work on real projects and to learn from experienced consultants. If you perform well during your internship and align with the firm's values and culture, you may be offered a full-time position upon graduation. My purely anecdotal observation is that in McKinsey this happens in around 80% of the cases.
It's always a good idea to make a positive impression during your internship and to let your team (leadership and EM) know that you are interested in a full-time position early on, plus discuss it with the other members of the team so that they can guide you along the way.
Ask for regular feedback sessions to understand where you stand and course-correct early.
Networking with other consultants and attending firm events can also help to increase your chances of being offered a full-time position. The more people know you, the easier it gets.