I was in Prague in June 2016 and noticed King Charles was really popular there, featured in all sorts of monuments and tourist attractions. At one point I saw a street exhibition titled Jaký byl? with explanations about reconstructions and the like of what he was like as a person.
Not really. Jaký je František is a question about his character, his size, his personality, etc. What is František is more of a question asking if František is a person or a dog, if he is a doctor or a mechanic. Those would be "Co je František" or "Čím je František" respectively.
If you know that František was not feeling well yesterday, and you want to ask how he is feeling today, how would you say "How is František today?"
I thought the same, however it was marked as correct! Though given the rest of the discussion here, it probably shouldn't be, as it has the sense of 'how is he feeling' rather than permanent characteristics.
Jak, jaký is how, correct? So why is How is František? incorrect. Is it not ok to say 'how is' when referring to a person's health or wellbeing
ion1122 has already answered your question. They are not the same.
Also, think of jaký, jaká, jaké (there are also plural forms) as adjectival words. So, when you ask about someone using those question words, in a way, an adjective is expected in answer. In general, anything to describe that person will be a valid response.
Things to remember:
jak is like an adverb, the answer is expected to qualify the verb (whatever the main verb in your question happens to be).
jaký is like an adjective (it has genders and plural forms), the answer is expected to qualify the person / thing that is the object of your question.