Terminology regarding missing symptom awareness depends on what is thought to be the cause. Anosognosia and other agnosias would be used for neurological disorders where self-monitoring is specifically impaired while denial, delusions and hallucinations would be used for psychiatric disorders. Denial could also be a psychiatric symptom concerning a somatic disorder. I'm not sure if other somatic fields than neurology have special terminology.
If a patient suffers from something that is not recognised as medical problem we call it hypochondria.
Not really. For example grief is not recognized as a medical problem, people suffer from it and we don't call it hypochondria.
Hypochondria is excessive worry about having a serious illness.
ETA: I think that whatever we choose to call a medical problem largely depends on our values and mere diversion from the biological norm does not a medical problem make. So the hypothetical patient could also simply disagree with others about what constitutes a medical problem.
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