Biden's use of "Bibi" is a reminder of just how long Netanyahu has been on the world stage, and more importantly how ubiquitous his presence in the U.S. has been during this period. Netanyahu has addressed Congress in his capacity as prime minister, testified before congressional committees as a private citizen, been a regular guest on American political talk shows, and made excellent use of his fluent and barely-accented English to give accessible interviews to U.S. media outlets. In short, Netanyahu is probably the world leader to whom Americans are most exposed.
This means that both American politicians and average Americans have a real sense of familiarity with Netanyahu, and in some cases may feel as if they know him intimately. Biden felt that he could refer to the Israeli prime minister by his informal nickname without any introduction or context on national television without having to worry that Americans watching at home would be confused about whom he was talking about. As soon as Biden referred to Bibi, everyone knew that the conversation had now switched to Israel. Netanyahu's constant presence in the U.S. and exposure to Americans gives him a degree of credibility with the public here by virtue of familiarity that other world leaders do not have, and that is a powerful tool that Netanyahu is able to wield.
But this familiarity is a double-edged sword. Biden and other politicians can casually refer to Bibi and it indicates a sense of comfort with the man, but it also signals that Netanyahu is not always taken seriously or at face value. Netanyahu has been warning that an Iranian bomb is imminent since the early 1990s, and there are many at this point who tune out his dire language given how long he has been using it. In September 2002, Netanyahu testified before the House Committee on Oversight and Government Reform that "there's no question" that Saddam Hussein had a nuclear program, adding, "If anyone makes an opposite assumption or cannot draw the lines connecting the dots, that is simply not an objective assessment of what has happened. Saddam is hell-bent on achieving atomic bombs, atomic capabilities, as soon as he can." As it turned out, Netanyahu's assessment was not exactly on the mark, and this episode added to the sense that his rhetoric does not always line up with the facts.
The infamous cartoon bomb at the United Nations General Assembly last month reinforced this notion that Netanyahu can be derided or mocked with ease. The infantilizing nickname that so many use when talking about him plays into the general idea that Netanyahu's warnings can be ignored or shunted aside, and that we have known him to play a certain character for so long that things he say should be taken with a grain of salt.
Finally, Biden's word choice says something larger about the current state of the U.S.-Israel relationship. In constantly referring to his friend "Bibi," Biden framed Israeli concerns over Iran almost entirely around Netanyahu - how close a relationship Biden has with the prime minister, how many times President Obama has met with or spoken with him. Doing so risks reducing the entire relationship with Israel to a personal one between Netanyahu and U.S. leaders, rather than the shared interests of both countries in countering a nuclear Iran.