No. They aren't saying that "age" sounds like 'h'. They're just close enough that any acoustic imperfections in the room (or the body of either person) could cause one to be misheard as the other.

A-g (age)

A-ch (H)

Ache doesn't sound like H, it sounds like ake. When we say age it sounds like eidg, which might sound somewhat similar to aitch.

Old person

The word 'age' kinda sounds like the letter H. Which is why the patient spelt put the rest of thier first name instead of stating hiw old they are.

Took me a minute to get, but figured I'd help others get the joke cause it got a chuckle out of me once I did.

Nobody. The patient misheard.

But why? Is there some kind of reference to something I'm missing?