That is correct, there are a lot of hp printers that use 564 ink cartridges, and a new variant the hp 564xl ink.
Why hp finally opted to use only one cartridge type for most of their printers is a manufacturing strategy to save on costs. Multiple cartridges complicates the manufacturing process, simply because there are constants that needs to be observed. Foremost of which is the machinery involved to do each type of cartridge which are expensive investments. Secondly, manpower has to be classified to man a particular model.
With only one cartridge to manufacture, supervision and management is much more straightforward. Besides with a single cartridge type, such could be served for printers that has gone out of manufacture. For instance, hp printers made prior to 2010 is not compatible to the recent ink cartridges, so other options must be tried on to keep printing.
It is more of a type that ‘fits all’ and it is intended to load recently made printers, such as the series 5400, the 5500 all in one (AIO), the 6300 AIO, the 6500 and 7500. It is also compatible with the PhotoSmart, Premium AIO, Pro and eAIO. The hp 564xl ink is a new variant added to the original HP 564.
The addition of a new variant is obviously to improve on the coverage in printed pages, because the hp 564xl ink cartridge contains more ink compared to the original. Ink cartridges are expensive per se, but if the ink volume is increased it will result in much tamer cost per page (CPP). It will hinder users from looking for alternative ink supplies.
In the comparison, a hp 564 ink cartridge ink capacity is placed around 7.5 ml, while the xl variant is good for 14 ml, or double the volume of the former. In page yield the smaller variant delivers 250 pages, while the xl variant is good up to 550 pages. Again over double the yield of the first cartridge.
In conclusion, hp printers that use HP 564 ink and the HP 564xl ink cartridges variant are many, so the decision to manufacture a lone cartridge model is a move in the right direction.