Puppy Linux is an old, mature distribution. Its ideas, principles and spirit have been well tested and well known for many years. It is therefore of no surprise to see other distributions that share the same principles and ideas. Some of them even behave very closely like Puppy.
These are distributions which are inspired in one way or another, from Puppy Linux. We consider them as a “derivative” in spirit, in idea. We call them as “offshots” to differentiate them from puplets, which is a true derivative.
Some of these offshots are forks of Puppy. Fatdog64, for example, was forked from Puppy Linux when it was at version 4.0. Although it is now an independently managed distribution and shares almost no code with the original Puppy it came from, it still claims to live up to Puppy’s ideals: small, fast, and versatile.
Some of the other offshots are inspired by the idea of Puppy Linux and adapts it for use in another distribution. DebianDog, for example, is an offshot of this type. It takes Debian Live-CD and molds it so it behaves very similarly with Puppy Linux.
Some takes a puplet and molds it into something that is more user friendly and called it something totally different (but acknowledges the source), for example, Simplicity Linux.
All of these distros are no longer considered as part of Puppy Linux distribution, but one can be mistaken to think that they are - because of their likeness and their similar behaviours. All of these distributions acknowledge that they owe their existence to Puppy Linux; if not directly in code, then in idea or in spirit.
Many of these distributions cross-share their improvements to each other; many of these distributions’ discussion are hosted in the same Puppy Linux forum (in different, distro-specific threads). Many of their developers and maintainers cross-post and comments on distributions different than their own. Many users of these distributions hop from one to another, and talk about them as if they are all Puppy.
Indeed, in spirit, they all are. All of these distribution developers and maintainers were Puppy Linux users in the past, who decided to improve it on their own, one way or another.
As such, it is just natural to consider them as part of the larger Puppy Linux community. This larger group is called as Puppy Linux family distributions: it includes all of the official/CE releases, puplets, and these offshots too.
If you have read it this far, you may also be interested to know more about various Puppy Linux distributions that have been released in the past, and their relationship to one another. Head on to Puppy Linux Family Tree to satisfy your curiosity.