The person who has said this might be meaning that religions have an identical, universal essence and differences among them are accidental, i.e. stemming from different cultural settings (that generate unique ritual forms and symbols for each religion), or interpretations of their followers, etc. And since it is these unique temporal aspects that people generally associate with the titles of each religion, and since it is also these aspects that obscure and mask the shared ontological origin of several religions, the negation of the said titles from their founders could be a way to draw attention to their identical esoteric aspect that can no longer be distinguished by these particular monikers.
As a side note, there's one prominent philosophical school that expounds on this idea, the School of Perennial Philosophy founded by Rene Guenon, a French intellectual who argued for transcendental unity of religions despite their differences in the domain of temporal manifestation. Guenon himself was a convert to Sufism and followed Islamic sharia since he believed that commitment to the external form of one particular religion is key in realizing the identical esoteric truths that has inspired different religious forms. Guenon studies these esoteric truths within the Neoplatonic philosophy which is based on the three ontological principles: The One, Intellect and Soul, (corresponding accordingly to the idea of God, angel and soul in religious terms.)
Based on the above insight, we can qualify the statement you quoted with respect to Prophet Muhammad. Prophet Muhammad was a Muslim in the sense that he submitted to the transcendent truths (e.g. God and angel of revelation) that inspired his own ministry but precisely because of the universal nature of these said truths, his message must no be regarded as particular concern of one particular group of people who specifically distinguish themselves as "Muslims" but of any human being who recognizes the universality of the said truths.