Most of the variations of the influenza viruses that infect humans cause one disease: seasonal influenza. Because the clinical disease caused by most influenza strains are indistinguishable, it makes no sense to consider them as separate diseases, any more than endocarditis secondary to MSSA should be considered a separate disease as endocarditis secondary to MRSA. In fact, since the treatment of influenza caused by different flu strains is the same (i.e. supportive care; Tamiflu is a glorified placebo), there's an even greater argument to consider endocarditis caused by bacteria with different resistance patterns as being different diseases (which no one would ever do).

I probably should have been clearer with "heart disease" in the sense that physicians don't even typically use the term "heart disease" in a formal sense. It's colloquially used as a synonym for ischemic heart disease (e.g. atherosclerosis of the coronary arteries leading to ischemia of the myocardium), but "ischemic heart disease" or "coronary artery disease" are the preferred terms in formal applications (e.g. scholarly publications, death certificates, hospital billing, etc...). But otherwise, we use much more specific terms. In other words, if a patient was being admitted for atrial fibrillation, "atrial fibrillation" is the admitting diagnosis, not "heart disease". Likewise, if a patient dies from myocarditis, "myocarditis" goes on the death certificate, not "heart disease". Using "heart disease" as an umbrella term to refer to all diseases that involve the heart renders the term useless (which perhaps it is anyway).