y1234 – Coming from a Black Hat Yeshivah background where we are accustomed to talking to Rabbanim in 3rd person I too was terribly bothered by this title “JB” and found it difficult to refer to him in that manner. However, commentator Joseph is correct. This title was not given to him by the Yeshivish world, but rather from his Talmidim. I don’t know details, so I’ll assume that commentator Joseph is right that he gave himself that title. If he did so it was probably to distinguish himself from his cousin Rav Berel Soloveichik z”l (both were named Yosef Dov after the Beis HaLevi). The Yeshivish world probably accepted this title used being that it helped distinguish which Rav Yosef Dov was being spoken about.
It should be noted that a lack of understanding of Kavod was unfortunately an issue in the early days in America. I was told by someone who was close to Rav Moshe Feinstein, z”l that his first talmidim used to refer to him as “the Chief”. I don’t believe that anyone meant to be disrespectful. They just didn’t grow up with the concept of speaking in 3rd person or giving illustrious titles.