Italics mine.
They know these genes? To what accuracy? For the many types of cancers?
[Side question on cancer(s): can/does each of the finite number of human cell types have a cancer unique to it? Or is it an organ thing only (which I don’t know if an “organ,” by definition, has a single cell type unique to it)?]
I seem to recall a National Lampoon comic about George Wallace giving his wife, Lurleen, cancer by feeding her meat that came from cancer-ridden rats left over from the cigarette studies.
They know that in some types of cancer, some genes are upregulated, and that in the more malignant, those genes are even more upregulated. And yes, they do know that some mutations in some genes can increase the cause of certain types of cancer (retinoblastoma, some leukemia (the Philadelphia mutation?), and breast cancer). There are probably others, but the above are some of the “classical” oncogenesis examples.
Organs (humans, animals) are composed of multple types of cells. Some of those cells are unique to that organ, but that doesn’t mean that they are the only type of cell found in that organ. For example, the subcutaneous (under the skin) tissue has fat and connective, fibrous tissue. You could have a tumor of the fat cells (lipoma/liposarcoma), or a tumor of the fibrous cell (fibroma/fibrosarcoma). And you can find fat cells and fibrous cells not just under the skin, but in and around most organs.
They only know some - just like for humans. And those genes don’t ensure cancer, they just heighten the risk of developing the cancer. With mice, you can breed lots of them, fast, so it isn’t a problem. With humans, not so much. There is constant ongoing research to identify genes that can be associated with particular cancers, but it is complex - sometimes it is a combination of genes providing increased risk, or a gene that interacts with an environmental factor (like a carcinogen or virus).
As for your side question - in some senses, all cancers are unique. They develop from a cell that has suffered multiple DNA mutations, and the changes they express affect the way the cancer develops. Again, researchers are beginning to sample and sequence the DNA from cancers to identify the critical changes in DNA relating to a particular cancer. However, there are also common failure modes across cancers - things like encapsulation (or not), the formation of blood supply to the tumour, rate of development, likelihood of spread. But not all cancers from a cell source can have all these features. Some do - skin cancers come in a wide range of types - basal cell carcinoma - generally slow, not very aggressive (just don’t leave it to erode your eyeball away - there is a thread somewhere about that). Compare that to another skin cancer like melanoma - aggressive, metastatic and far more dangerous. But the spreading cancer can carry the nature of the source with it.
And KarlGrenze has a better answer than this, but I typed it, so …
And just to point out, melanoma is a cancer of melanocytes, the pigment-producing cells, which are different from the basal cells that compose basal cell carcinoma. Basal cells are the regenerative cells of things like the epidermis (upper layer of skin, and there they form the stratum basale), hair follicles, and some glands, which shed their surface cells constantly.
Transgenic studies are also done in mice (inserting genes), and they have found homologues between at least some of the human genes and the mouse genes, so they know if they chop/add/mutate/remove/do magic on x part of the mouse genome, it corresponds to y part in the human genome.
There are, as noted above, viral caused cancer. Such as cervical caused by papilloma virus.But I have no reason to think that the cancer is contagious, although the virus is.
When I was a student working in a lab there was something called ascites tumor that could be transmitted from mouse to mouse by extracting from the abdominal cavity of one mouse (where it grew) and injecting it another. That doesn’t make it contagious in the usual sense and I don’t think it was.
Well, the virus that causes the cancer is contagious. 
But the infection would just transmit the virus which will replicate in the new host and cause cancer, not the whole tumor cells.
There are, AFAIK, two parasitic tumors known in animals. The above mentioned Tasmanian Devil Facial Tumor, and a very benign, easily treatable, venereal tumor of dogs.
Bandicoots, which are also marsupials, do have a cancer-causing virus (either papilloma or herpesvirus, can’t remember off the top of my head). Cancer-causing viruses have also been found in all domestic species, some marine mammals, and some hooved wildlife.