It's totally possible and it already exists. You train your immune system (that's the complicated part, you have to feed it some part of cancer cells and make it understan that's the bad guys. Which is very complicated) to fight the cancerous cells.

Let's not firget there are around 300 families of cancer and there isn't any vaccine against all of them obviously.

That said, from russia it is obvious bs.

Would it even be classified as a vaccine since cancer isn’t a virus?

Increasingly everything injected gets called that in popular media, it seems.

The article says he didn't specify what kinds of cancer he's talking about, or any exact timeline. You might be able to prevent a cancer, but all cancers seems pretty impossible, short of hypothetical nanobots that turn you into a disease-immune superhuman.

Is a cancer vaccine even possible as a concept?

Yes. A recently-found example of cancer resistance is wolves in Chernobyl.

Would it even be classified as a vaccine since cancer isn't a virus?

Yes, and some oncovaccines are already approved.

There are some types of cancer vaccine, but from what I know they're usually given to people who already have cancer. A college classmate of mine told me he had a bladder cancer vaccine

I'm especially ignorant on cancer medicine, but I do know that Cuba has developed a vaccine that is supposed to prevent a type of lung cancer. My guess would be a similar sort of result here

Cuba has had a lung cancer vaccine now for about a decade. The US is also testing mRNA cancer vaccines currently.