When students enter higher education, they not only start learning and studying but begin a journey of becoming someone new in relation to themselves and to society. Scholarly research has increasingly emphasised this transformative element of higher education yet, to date, the role of assessment has received little attention in the processes of being and becoming. This is a crucial limitation given how profoundly assessment - exams, self- and peer-assessment, grades, rankings, metrics - characterises contemporary higher education. In this conceptual study, we propose a novel way of rethinking assessment as a matter of being and becoming. We bring together the separate fields of research on assessment and the philosophy of higher education by theorising other-formation and self-formation in and through assessment encounters. First, we discuss how assessment shapes students in 'the measured university' ("other-formation"). We then reconceptualise assessment from the viewpoint of how it promotes student agency and self-reflexivity ("self-formation"). Finally, we discuss some frequently used assessment practices in light of student self/other-formation. Our study sets the agenda of understanding student formation in and through assessment by laying the philosophical foundation for future studies.