Cancer is not the only disease to which a definite article has been attached. The malady caused by the bacterium Yersinia pestis is often referred to as the plague, and one of its other descriptors, the Black Death, is another example.
My sense is that one of the reasons for attaching a definite article to the name of a disease may be that it reflects the notion people have of it as a mysterious and fated phenomenon ("the big C"); an irresistible and usually fatal affliction against which people are powerless, either because it is their destiny to succumb to it and/or because there is no known or guaranteed antidote or remedy.
However, that's only an unsubstantiated hypothesis, and it doesn't explain why not every much-feared and hard-to-cure or -prevent disease has acquired the definite article.
The likelihood of a Scandinavian origin for the use of the definite article when referring to cancer is not very plausible, for a couple of different reasons:
1) Among the main Scandinavian languages, only in Swedish is the usual word for cancer 'cancer':
- Danish (DA): kræft, with cancer being a parallel but less-often-used term
- Norwegian (NO): kreft
- Swedish (SV): cancer
2) In most Scandinavian languages, the definite form of a noun is indicated by the addition of a suffix (typically -en or -et; e.g. the Danish word for liver [the organ] is 'lever', and the liver is 'leveren'). But when someone is described as suffering from cancer -- or indeed any disease -- this form is not the one used.
For example, He has liver cancer would be rendered as follows (the literal translation in all cases is "He has cancer in the liver"):
DA: Han har kræft/cancer i leveren
NO: Han har kreft i levern
SV: Han har cancer i levern
The bald description He is suffering from cancer follows the same pattern (in all cases it translates literally as "He suffers of cancer"):
DA: Han lider af kræft/cancer
NO: Han lider av kreft
SV: Han lider av cancer