It's a metamorphosed Axolotl, and yes, these do very much count as true salamanders. Axolotls don't metamorphose in the wild. The ability to reproduce as a sexually precocious larvae evolved (devolved?) as a way to speed up reproduction and avoid an unnecessary reliance on dry land. It is extremely rare for Axolotls to spontaneously metamorphose in captivity, but using hormones it's pretty easily induced. Most people do it in a bad way (I'm not going to describe good and bad methods for risk of encouraging people to do it) and the metamorphosed animals typically have poor health and don't live for long. If done properly they can be healthier and live longer, but still won't thrive as well as natural neotenic larvae. The salamanders are also pretty boring pets compared to the natural type.
Axolotls are legal to keep and breed in Australia. Not so long ago you could keep other amphibians, but Axolotls are now the only one.