Why is the bullet cluster's name not the only one?
Most astronomical sources don't have a poetic name such as the Bullet Cluster. Such names are usually coined for objects that may have a particular interest to the public (or the astronomical community), e.g. for their beauty or extraordinary physical processes going on.
In addition to their poetic names, objects typically have one or more other names, depending on which catalogue they appear in. An example is Betelgeuse, which is also called "α Ori" (being the brightest star in Orion), HR 2061 in the Bright Star Catalogue, HD 39801 in the Henry Draper Catalogue, and CCDM J05552+0724 in the Catalog of Components of Double & Multiple stars .
The name typically reflects coordinates
Sometimes, as in the HR and HD case, the number simply refers to the number in a list. Other times, as in the case of the CCDM, the number refers to the coordinates on the sky. And in that case, the name is not completely unambiguous, since coordinates may be given with more or less precision. In the case of Betelgeuse, "J05552+0724" means that its right ascension is 05h and 55.2m, and its declination is 7º and 24m. More precisely, however, the coordinates of Betelgeuse are R.A. = 05h 55m 10.30536s, dec. = +07º 24' 25.4304", so in the CCDM catalogue you could also get around calling it "J0555172+0724424".
In your case, the numbers also refer to the coordinates with varying precision, which are R.A. = 06h 58m 37.9s, dec. = –55º 57' 0". The cluster appears in various catalogues of galaxy clusters, X-ray sources, and weak gravitational lenses, leading to its different names.
Coordinates depend on the wavelength
In addition to the varying precision, the coordinates of extended objects such as the Bullet Cluster are not well-defined. Depending on the type of object, the coordinates are typically defined as the maximum of the surface brightness, but could also be a region roughly in the middle of multiple peaks. However, the surface brightness of an object doesn't necessarily peak in the same location across all wavelengths. Hence, and object could have different coordinate-inspired names in catalogues of different wavelength observations.
Almost every object in space outside the solar system has multiple names (identifiers) because different astronomical surveys use different naming conventions. The Bullet Cluster has at least 25 different identifiers per the Set of Identifications, Measurements and Bibliography for Astronomical Data(SIMBAD).