I wonder if the use of 'ottoman' as 'footrest' in English has a similar origin story.

Showing the soles of your feet to someone is extremely disrespectful in many middle eastern cultures including the Ottoman culture.

Not really; it was more along the lines of "Ottoman-style seating" being referred to as "an ottoman", in the same way people refer to "a long chair" (chaise longue) as a "chaise".