Cancer is like a bacterial infection, but with cells from the host. Of course if your lung has an infection of colon cells those don’t suddenly become lung cells.
I don’t see how that makes it different diseases. It is only treated that way because we aren’t attacking the root cause (i.e. the actual broken replication inhibitor in whatever line of cells).
https://www.nature.com/articles/s41586-020-1969-6
Figure 2 shows the different mutations present in cells in a variety of cancers carry. These genes have a variety of functions. The DNA of cancer cells from different tumor types is different, and TP53, the gene featured in OP, is never mutated in some cancer types. In the end the discussion is a little semantic (what defines a different disease), but if the prognosis, treatment, and molecular mechanisms behind different types of cancers are different, I would consider it fair to consider them different diseases.
For the cancer to not get killed immediately several things have to get broken, not just the replication inhibitor. Once they get to that state is not far fetched to think that they can have or get in the future different mutations, not specifically required to get cancer in the first place.
The comments below us talk about the folate receptor mutation that not all cancers have.