For the cells to act exactly as described would require cooperation between different metastasising cell lines, both during infection and when growing structures in the new host. That implies a kind of control not usually found in a tumour.
Even a simple tumour tends to grow into an amorphous mass unless it's growing from stem cells, in which case it can form recognisable structures including multiple cell types. If using a single line involving stem cells is acceptable for an answer, then I can think of two ways it might work.
1. Take a stem cell and make it cancerous
In the comments, Jordan suggested a stem cell becoming cancerous and some of its daughter cells developing into limbs. There are tumours called teratomas that have been found containing skin, hair, eyes and teeth, so that part at least is possible. Teratomas are believed to originate from germ cells (which are the ultimate stem cells) and do not usually metastasise.
To become transmissible the stem cell would have to become more mobile than usual as well as becoming cancerous, and would have to wait until it had infected a new host before it started to differentiate. That is three changes that need to be made to the cell's behaviour, all without losing its ability to act like a stem cell - and "becoming more mobile" might involve activating multiple genes.
2. Take a cancerous cell and make it stem cell-like
There is evidence that some cancer cells can become partly de-differentiated, turning "back" into more potent stem cells. This raises the question of whether a transmissible tumour cell could become so dedifferentiated that it could give rise to an ear or whatever. A normal(ish) cell can be made into a fully 'pluripotent' stem cell in the lab, but it requires a set of changes that don't normally happen in an animal.
Dedifferentiation means losing most of the cell's differentiated characteristics, and my (non-expert) guess is that this would include the ability to infect a new host.
For the tumour to be infections and grow new structures, the same kind of dedifferentiation would have to happen again in each new host. That implies that either this tumour has cells that frequently dedifferentiate spontaneously, or they do it in response to certain conditions in the host.
Both of the above scenarios require several unusual changes to a cell. Tumour cells divide and mutate frequently so any one change is likely to happen from time to time, but getting a cell line to acquire them all while still being viable would be a bit like building a working radio from scratch - possible with a lot of work and the right technology, but unlikely by chance.
In this case the "right technology" might be available soon if it isn't already. Recent years have seen a lot of interest in stem cell generation, including its implications for tissue regeneration and cancer. Whether anyone would or should actually try to produce a transmissible cancer that can create complex structures is another matter.
If it did happen by chance, the new cancer would suddenly be growing new structures in its host that are likely to cause more problems than a normal tumour, so it might have trouble spreading through the population.
Every cell type in the new tumour/organ/limb will carry a (slightly?) different set of antigens, so I'd expect that having more cell types involved means getting more immune rejection. Using inbred hosts would be the way to go.